Clear[g,ω,d,n,r,rcheck,q]

g=haagerupFamilyBigraph[7]
ω=-1
d=DimensionOfGenerator[g]
n=DepthOfBranchPoint[g]
r=1
rcheck
q=qFromd[d]

S={d,n,ω,r,rcheck}

Assert[product[STrain[GeneratorData@@S,{1}], f[2n+2], STrain[S,{1}]]==((qInteger[n+1][q]qInteger[2n+2][q])/(qInteger[n][q]qInteger[n+2][q]))]

GeneratorData@@S

Assert[product[STrain[GeneratorData@@S,{1,1}], f[2n+4], STrain[S,{1,1}]]==((((qInteger[n+1][q])^2)qInteger[2n+2][q])/(qInteger[2][q] ((qInteger[n][q])^2)qInteger[2n+3][q]))(qInteger[n+3][q]-qInteger[2][q]qInteger[n][q])]

RootReduce[expandJWRight[product[STrain[GeneratorData@@S,{1}], f[2n+2], STrain[GeneratorData@@S,{1}]]]/.{qi[s_]->qInteger[s][q]}]

RootReduce[expandJWRight[product[STrain[GeneratorData@@S,{1,1}], f[2n+4], STrain[GeneratorData@@S,{1,1}]]]/.{qi[s_]->qInteger[s][q]}]