This is for comparison purposes -- it looks at the actual low weight eigenspace for Haagerup.

FusionAtlasPaths={"~/projects/fusionatlas/code/package/"};
$Path=$Path~Join~FusionAtlasPaths;
<<FusionAtlas`

SetDebugMode[True]

g=HaagerupBigraph

LowestWeightEigenspace[g,-1]=.

matrices=Cases[LowestWeightEigenspace[g,-1][c][[-1]],{{Vertex[0,1],Vertex[DepthOfBranchPoint[g]+1,_]},_}|{{Vertex[2,1],Vertex[d_/;DepthOfBranchPoint[g]-1≤d≤DepthOfBranchPoint[g]+1,_]},_}];

matrices/.{i:{__Vertex},m_}:>{i,MatrixForm[m]}

Union[Cases[matrices,_c,∞]]

moment[k_/;k≥1][0]:=Plus@@(Cases[matrices,{{Vertex[0,1],_},_}]/.{i:{__Vertex},m_}:>(w[0]@@i)Tr[MatrixPower[m,k]])

moment[k_/;k≥1][1]:=Plus@@(matrices/.{i:{__Vertex},m_}:>(w[1]@@i)Tr[MatrixPower[m,k][[slice@@i,slice@@i]]])

slice[v1_,v2_]:=Flatten[Position[FusionAtlas`GraphPlanarAlgebra`Private`GraphPathsBetween[g,DepthOfBranchPoint[g]+1,v1,v2],{_,_,Vertex[1,1],__}]]

w[0][v1_,v2_]:=((LopsidedDimension[g,v2])/(LopsidedDimension[g,v1]))

w[1][v1_,v2_]:=((LopsidedDimension[g,v1]LopsidedDimension[g,v2])/((LopsidedDimension[g,Vertex[1,1]])^2))

Collect[moment[1][0],_c,Together]

Collect[moment[1][1],_c,Together]

Collect[moment[2][0],_c,Together]

Collect[moment[2][1],_c,Together]

index=ToNumberField[RootReduce[((DimensionOfGenerator[g])^2)],NumberFieldGenerator[g]]

conditions=S2Conditions[g,-1][c];

solution=SolveQuadratics[conditions,{},Union[Cases[conditions,_c,∞]],NumberFieldGenerator[g]]

solution[[1,1]]

matrices2=matrices/.solution[[4,1]];

matrices2/.{i:{__Vertex},m_}:>{i,MatrixForm[m]}

N@matrices2[[4,2]]//MatrixForm

matrices=matrices2;

Collect[index moment[2][0]-moment[2][1],_c,Together]/.solution[[1,1]]

Collect[index moment[3][0]-moment[3][1],_c,Together]/.solution[[1,1]]

Collect[index moment[4][0]-moment[4][1],_c,Together]/.solution[[1,1]]

Collect[index moment[6][0]-moment[6][1],_c,Together]/.solution[[1,1]]

matrices