WARNING: Here, I'm assuming epsilon is between 1/4 and 1/3.  A different decomposition is needed for bigger epsilon!
I'm also restricting polynomials to only involve two  of the three variables at a time.

VariableNames[number_,string_]:=Table[ToExpression[ToString[string]<>ToString[i]],{i,1,number}]

Mon[x_,y_,z_,S_]:=(x^(S[[1]]))*(y^(S[[2]]))*(z^(S[[3]]))

PolyMon[x_,y_,z_,S_]:=Module[{permset},permset=Permutations[S];Sum[Mon[x,y,z,permset[[i]]],{i,1,Length[permset]}]]

PossibleSignatures[d_]:=Module[{i,j},Union[Flatten[Union[Table[Table[Append[IntegerPartitions[i,{2}][[j]],0],{j,1,Length[IntegerPartitions[i,{2}]]}],{i,0,d}]],1],
Flatten[Union[Table[Table[Append[Append[IntegerPartitions[i,{1}][[j]],0],0],{j,1,Length[IntegerPartitions[i,{1}]]}],{i,0,d}]],1],
Flatten[Union[Table[Table[Append[Append[Append[IntegerPartitions[i,{0}][[j]],0],0],0],{j,1,Length[IntegerPartitions[i,{0}]]}],{i,0,d}]],1]]]

PolyGen[x_,y_,z_,d_,VarString_]:=Module[{types,vars},types=Flatten[Map[Permutations,PossibleSignatures[d]],1];vars=VariableNames[Length[types],VarString];{Sum[vars[[i]]*Mon[x,y,z,types[[i]]],{i,1,Length[types]}],vars}]

PolyInt[Expr_,Var_,Bnds_]:=Module[{n,tmp},tmp=Expand[Expr];Sum[Coefficient[tmp,Var,n]*(Bnds[[2]]^(n+1)-Bnds[[1]]^(n+1))/(n+1),{n,0,Max[0,Exponent[tmp,Var]]}]]

PolytopeInt[Expr_,Polytope_]:=Module[{tmp},If[Length[Polytope]==0,Expr,tmp=PolyInt[Expr,Polytope[[1]][[1]],Polytope[[1]][[2]]];
PolytopeInt[tmp,Delete[Polytope,1]]]]

I1[x_,y_,z_,eps_]:={{{z,{x,1-eps-x}},{y,{0,x}},{x,{0,1/2-eps/2}}}};
I2[x_,y_,z_,eps_]:={{{y,{0,1-eps-z}},{x,{1-eps-z,z}},{z,{1/2-eps/2,1/2+eps/2}}}};
I3[x_,y_,z_,eps_]:={{{y,{0,1-eps-z}},{x,{1-eps-z,1+eps-z}},{z,{1/2+eps/2,1-eps}}}};
I4[x_,y_,z_,eps_]:={{{z,{1-eps-y,1+eps-x}},{x,{y,y+2eps}},{y,{0,1/2-3eps/2}}}};
I5[x_,y_,z_,eps_]:={{{z,{1-eps-y,1+eps-x}},{x,{y,1-eps-y}},{y,{1/2-3eps/2,1/2-eps}}}};
I6[x_,y_,z_,eps_]:={{{z,{1-eps-y,3/2-x-y}},{x,{y,1-eps-y}},{y,{1/2-eps,1/2-eps/2}}}};
I7[x_,y_,z_,eps_]:={{{y,{0,1-eps-z}},{x,{1+eps-z,z}},{z,{1/2+eps/2,1-eps}}}};
I8[x_,y_,z_,eps_]:={{{x,{1+eps-z,1-eps-y}},{z,{1-eps-y,1/2+eps}},{y,{0,1/2-3eps/2}}}};
I9[x_,y_,z_,eps_]:={{{x,{1+eps-z,1-eps-y}},{z,{y+2eps,1/2+eps}},{y,{1/2-3eps/2,1/2-eps}}}};
I10[x_,y_,z_,eps_]:={{{y,{0,3/2-x-z}},{x,{1+eps-z,3/2-z}},{z,{1/2+eps,1/2+2eps}}}};
I11[x_,y_,z_,eps_]:={{{y,{0,3/2-x-z}},{x,{1/2-eps,3/2-z}},{z,{1/2+2eps,1+eps}}}};
I12[x_,y_,z_,eps_]:={{{z,{1+eps-x,1+eps-y}},{y,{0,x}},{x,{0,1/2-eps}}}};
I13[x_,y_,z_,eps_]:={{{z,{1+eps-y,3/2-x-y}},{y,{0,x}},{x,{0,1/2-eps}}}};
I14[x_,y_,z_,eps_]:={{{z,{x,3/2-x-y}},{y,{1-eps-x,3/2-2x}},{x,{1/2+eps/2,1-eps}}}};
I15[x_,y_,z_,eps_]:={{{z,{x,3/2-x-y}},{y,{0,3/2-2x}},{x,{1-eps,3/4}}}};
I16[x_,y_,z_,eps_]:={{{z,{1+eps-x,3/2-x-y}},{y,{1-eps-x,1/2-eps}},{x,{1/2,1/2+eps/2}}}};

IIntegral[Aguy_,Bguy_,Cguy_,Eguy_,Sguy_,Tguy_,Uguy_,Gguy_,Hguy_,eps_]:=Module[{},
Iterm1=PolytopeInt[(Aguy^2),First[I1[x,y,z,eps]]];
Iterm2=PolytopeInt[(Bguy^2),First[I2[x,y,z,eps]]];
Iterm3=PolytopeInt[(Bguy^2),First[I3[x,y,z,eps]]];
Iterm4=PolytopeInt[(Cguy^2),First[I4[x,y,z,eps]]];
Iterm5=PolytopeInt[(Cguy^2),First[I5[x,y,z,eps]]];
Iterm6=PolytopeInt[(Cguy^2),First[I6[x,y,z,eps]]];
Iterm7=PolytopeInt[(Eguy^2),First[I7[x,y,z,eps]]];
Iterm8=PolytopeInt[(Sguy^2),First[I8[x,y,z,eps]]];
Iterm9=PolytopeInt[(Sguy^2),First[I9[x,y,z,eps]]];
Iterm10=PolytopeInt[(Tguy^2),First[I10[x,y,z,eps]]];
Iterm11=PolytopeInt[(Tguy^2),First[I11[x,y,z,eps]]];
Iterm12=PolytopeInt[(Uguy^2),First[I12[x,y,z,eps]]];
Iterm13=PolytopeInt[(Gguy^2),First[I13[x,y,z,eps]]];
Iterm14=PolytopeInt[(Hguy^2),First[I14[x,y,z,eps]]];
Iterm15=PolytopeInt[(Hguy^2),First[I15[x,y,z,eps]]];
Iterm16=PolytopeInt[(Hguy^2),First[I16[x,y,z,eps]]];
Expand[6*(Iterm1+Iterm2+Iterm3+Iterm4+Iterm5+Iterm6+Iterm7+Iterm8+Iterm9+Iterm10+Iterm11+Iterm12+Iterm13+Iterm14+Iterm15+Iterm16)]
]

J1[x_,y_,eps_]:={{{y,{0,x}},{x,{0,1/2-eps}}}};
J2[x_,y_,eps_]:={{{y,{0,1/2-eps}},{x,{1/2-eps,1/2-eps/2}}}};
J3[x_,y_,eps_]:={{{y,{1/2-eps,x}},{x,{1/2-eps,1/2-eps/2}}}};
J4[x_,y_,eps_]:={{{y,{1/2-eps,1-eps-x}},{x,{1/2-eps/2,1/2}}}};
J5[x_,y_,eps_]:={{{y,{0,1/2-eps}},{x,{1/2-eps/2,1/2}}}};
J6[x_,y_,eps_]:={{{y,{0,1-eps-x}},{x,{1/2,2eps}}}};
J7[x_,y_,eps_]:={{{y,{x-2eps,1-eps-x}},{x,{2eps,1/2+eps/2}}}};
J8[x_,y_,eps_]:={{{y,{0,x-2eps}},{x,{2eps,1/2+eps/2}}}};
J9[x_,y_,eps_]:={{{y,{0,1-eps-x}},{x,{1/2+eps/2,1-eps}}}};

JIntegral[Axyzguy_,Azyxguy_,Ayzxguy_,Bxyzguy_,Bzyxguy_,Cxyzguy_,Exyzguy_,Ezyxguy_,Sxyzguy_,Txyzguy_,Uxyzguy_,Gxyzguy_,eps_]:=Module[{},
Jterm1=PolytopeInt[(((PolyInt[Ayzxguy,z,{0,y}]+PolyInt[Azyxguy,z,{y,x}]+PolyInt[Axyzguy,z,{x,1-eps-x}]+PolyInt[Bxyzguy,z,{1-eps-x,1-eps-y}]+PolyInt[Cxyzguy,z,{1-eps-y,1+eps-x}]+PolyInt[Uxyzguy,z,{1+eps-x,1+eps-y}]+PolyInt[Gxyzguy,z,{1+eps-y,3/2-x-y}]))^2),First[J1[x,y,eps]]];
Jterm2=PolytopeInt[(((PolyInt[Ayzxguy,z,{0,y}]+PolyInt[Azyxguy,z,{y,x}]+PolyInt[Axyzguy,z,{x,1-eps-x}]+PolyInt[Bxyzguy,z,{1-eps-x,1-eps-y}]+PolyInt[Cxyzguy,z,{1-eps-y,1+eps-x}]+PolyInt[Txyzguy,z,{1+eps-x,3/2-x-y}]))^2),First[J2[x,y,eps]]];
Jterm3=PolytopeInt[(((PolyInt[Ayzxguy,z,{0,y}]+PolyInt[Azyxguy,z,{y,x}]+PolyInt[Axyzguy,z,{x,1-eps-x}]+PolyInt[Bxyzguy,z,{1-eps-x,1-eps-y}]+PolyInt[Cxyzguy,z,{1-eps-y,3/2-x-y}]))^2),First[J3[x,y,eps]]];
Jterm4=PolytopeInt[(((PolyInt[Ayzxguy,z,{0,y}]+PolyInt[Azyxguy,z,{y,1-eps-x}]+PolyInt[Bzyxguy,z,{1-eps-x,x}]+PolyInt[Bxyzguy,z,{x,1-eps-y}]+PolyInt[Cxyzguy,z,{1-eps-y,3/2-x-y}]))^2),First[J4[x,y,eps]]];
Jterm5=PolytopeInt[(((PolyInt[Ayzxguy,z,{0,y}]+PolyInt[Azyxguy,z,{y,1-eps-x}]+PolyInt[Bzyxguy,z,{1-eps-x,x}]+PolyInt[Bxyzguy,z,{x,1-eps-y}]+PolyInt[Cxyzguy,z,{1-eps-y,1+eps-x}]+PolyInt[Txyzguy,z,{1+eps-x,3/2-x-y}]))^2),First[J5[x,y,eps]]];
Jterm6=PolytopeInt[(((PolyInt[Ayzxguy,z,{0,y}]+PolyInt[Azyxguy,z,{y,1-eps-x}]+PolyInt[Bzyxguy,z,{1-eps-x,x}]+PolyInt[Bxyzguy,z,{x,1-eps-y}]+PolyInt[Cxyzguy,z,{1-eps-y,1+eps-x}]+PolyInt[Sxyzguy,z,{1+eps-x,1/2+eps}]+PolyInt[Txyzguy,z,{1/2+eps,3/2-x-y}]))^2),First[J6[x,y,eps]]];
Jterm7=PolytopeInt[(((PolyInt[Ayzxguy,z,{0,y}]+PolyInt[Azyxguy,z,{y,1-eps-x}]+PolyInt[Bzyxguy,z,{1-eps-x,x}]+PolyInt[Bxyzguy,z,{x,1-eps-y}]+PolyInt[Cxyzguy,z,{1-eps-y,1+eps-x}]+PolyInt[Sxyzguy,z,{1+eps-x,1/2+eps}]+PolyInt[Txyzguy,z,{1/2+eps,3/2-x-y}]))^2),First[J7[x,y,eps]]];
Jterm8=PolytopeInt[(((PolyInt[Ayzxguy,z,{0,y}]+PolyInt[Azyxguy,z,{y,1-eps-x}]+PolyInt[Bzyxguy,z,{1-eps-x,x}]+PolyInt[Bxyzguy,z,{x,1+eps-x}]+PolyInt[Exyzguy,z,{1+eps-x,1-eps-y}]+PolyInt[Sxyzguy,z,{1-eps-y,1/2+eps}]+PolyInt[Txyzguy,z,{1/2+eps,3/2-x-y}]))^2),First[J8[x,y,eps]]];
Jterm9=PolytopeInt[(((PolyInt[Ayzxguy,z,{0,y}]+PolyInt[Azyxguy,z,{y,1-eps-x}]+PolyInt[Bzyxguy,z,{1-eps-x,1+eps-x}]+PolyInt[Ezyxguy,z,{1+eps-x,x}]+PolyInt[Exyzguy,z,{x,1-eps-y}]+PolyInt[Sxyzguy,z,{1-eps-y,1/2+eps}]+PolyInt[Txyzguy,z,{1/2+eps,3/2-x-y}]))^2),First[J9[x,y,eps]]];
Expand[6*(Jterm1+Jterm2+Jterm3+Jterm4+Jterm5+Jterm6+Jterm7+Jterm8+Jterm9)]
]

NewAnswer[d_,eps_]:=Module[{vars,SetOfConditions,SetOfReductions,count,Axyz,Ayzx,Azyx,Bxyz,Bzyx,Cxyz,Exyz,Eyzx,Ezyx,Sxyz,Syzx,Txyz,Tyzx,Uxyz,Uyzx,Gxyz,Gyzx,Gzyx,Hxyz,Hyzx,AAvars,BBvars,CCvars,EEvars,SSvars,TTvars,UUvars,GGvars,HHvars},
Print[DateList[]];
{Axyz,AAvars}=PolyGen[x,y,z,2,AA];
Ayzx=First[PolyGen[y,z,x,2,AA]];
Azyx=First[PolyGen[z,y,x,2,AA]];
{Bxyz,BBvars}=PolyGen[x,y,z,2,BB];
Bzyx=First[PolyGen[z,y,x,2,BB]];
{Cxyz,CCvars}=PolyGen[x,y,z,2,CC];
{Exyz,EEvars}=PolyGen[x,y,z,d,EE];
Eyzx=First[PolyGen[y,z,x,d,EE]];
Ezyx=First[PolyGen[z,y,x,d,EE]];
{Sxyz,SSvars}=PolyGen[x,y,z,d,SS];
Syzx=First[PolyGen[y,z,x,d,SS]];
{Txyz,TTvars}=PolyGen[x,y,z,d,TT];
Tyzx=First[PolyGen[y,z,x,d,TT]];
{Uxyz,UUvars}=PolyGen[x,y,z,d,UU];
Uyzx=First[PolyGen[y,z,x,d,UU]];
{Gxyz,GGvars}=PolyGen[x,y,z,d,GG];
Gyzx=First[PolyGen[y,z,x,d,GG]];
Gzyx=First[PolyGen[z,y,x,d,GG]];
{Hxyz,HHvars}=PolyGen[x,y,z,d,HH];
Hyzx=First[PolyGen[y,z,x,d,HH]];
vars=Union[AAvars,BBvars,CCvars,EEvars,SSvars,TTvars,UUvars,GGvars,HHvars];
SetOfConditions=Union[Flatten[CoefficientArrays[PolyInt[Gyzx,z,{0,3/2-x-y}],{x,y}]//Normal],Flatten[CoefficientArrays[PolyInt[Gyzx,z,{0,y}]+PolyInt[Gzyx,z,{y,3/2-x-y}],{x,y}]//Normal],Flatten[CoefficientArrays[PolyInt[Uyzx,z,{0,1+eps-x}]+PolyInt[Gyzx,z,{1+eps-x,y}]+PolyInt[Gzyx,z,{y,3/2-x-y}],{x,y}]//Normal],Flatten[CoefficientArrays[PolyInt[Uyzx,z,{0,1+eps-x}]+PolyInt[Gyzx,z,{1+eps-x,3/2-x-y}],{x,y}]//Normal],Flatten[CoefficientArrays[PolyInt[Tyzx,z,{0,3/2-x-y}],{x,y}]//Normal],Flatten[CoefficientArrays[PolyInt[Syzx,z,{0,1-eps-y}]+PolyInt[Hyzx,z,{1-eps-y,3/2-x-y}],{x,y}]//Normal],Flatten[CoefficientArrays[PolyInt[Eyzx,z,{0,1-eps-x}]+PolyInt[Syzx,z,{1-eps-x,1-eps-y}]+PolyInt[Hyzx,z,{1-eps-y,3/2-x-y}],{x,y}]//Normal],Flatten[CoefficientArrays[PolyInt[Hyzx,z,{0,3/2-x-y}],{x,y}]//Normal]];
SetOfReductions=Reduce[SetOfConditions==Table[0,{i,1,Length[SetOfConditions]}]];
count=1;
While[count≤Length[SetOfReductions],
vars=Drop[vars,First[Position[vars,SetOfReductions[[count,1]]]]];
Axyz=Axyz/.SetOfReductions[[count,1]]->SetOfReductions[[count,2]];
Ayzx=Ayzx/.SetOfReductions[[count,1]]->SetOfReductions[[count,2]];
Azyx=Azyx/.SetOfReductions[[count,1]]->SetOfReductions[[count,2]];
Bxyz=Bxyz/.SetOfReductions[[count,1]]->SetOfReductions[[count,2]];
Bzyx=Bzyx/.SetOfReductions[[count,1]]->SetOfReductions[[count,2]];
Cxyz=Cxyz/.SetOfReductions[[count,1]]->SetOfReductions[[count,2]];
Exyz=Exyz/.SetOfReductions[[count,1]]->SetOfReductions[[count,2]];
Ezyx=Ezyx/.SetOfReductions[[count,1]]->SetOfReductions[[count,2]];
Sxyz=Sxyz/.SetOfReductions[[count,1]]->SetOfReductions[[count,2]];
Txyz=Txyz/.SetOfReductions[[count,1]]->SetOfReductions[[count,2]];
Uxyz=Uxyz/.SetOfReductions[[count,1]]->SetOfReductions[[count,2]];
Gxyz=Gxyz/.SetOfReductions[[count,1]]->SetOfReductions[[count,2]];
Hxyz=Hxyz/.SetOfReductions[[count,1]]->SetOfReductions[[count,2]];
count=count+1
];
Iint=IIntegral[Axyz,Bxyz,Cxyz,Exyz,Sxyz,Txyz,Uxyz,Gxyz,Hxyz,eps];
Print["I integral done ",DateList[]];
Jint=JIntegral[Axyz,Azyx,Ayzx,Bxyz,Bzyx,Cxyz,Exyz,Ezyx,Sxyz,Txyz,Uxyz,Gxyz,eps];
Print["J integral done ",DateList[]];
Print["Here are the variables ",vars];
M1=CoefficientArrays[Expand[Iint],vars,Symmetric->True][[3]];
M2=CoefficientArrays[Expand[Jint],vars,Symmetric->True][[3]];
M3=Inverse[M1].M2;
Print["Inverse matrix done ",DateList[]];
RatVec=Rationalize[Eigenvectors[N[M3,150],1][[1]],10^(-40)];
Print["Vector found ",DateList[]];
Ratio=(RatVec.M2.RatVec)/(RatVec.M1.RatVec);
Clear["AA*","BB*","CC*","EE*","SS*","TT1*","TT2*","TT3*","TT3*","TT4*","TT5*","TT6*","TT7*","TT8*","TT9*","UU*","GG*","HH*"];
Print["Ratio is ",N[Ratio,20]];
N[Ratio,20]
]

NewAnswer[5,1/4]