Unbounded join defect for nice finite groups

Determine whether the join defect \(\delta_G(X,Y)=\operatorname{ind}_G X+\operatorname{ind}_G Y+1-\operatorname{ind}_G(X*Y)\) is unbounded as X and Y range over finite free G-simplicial complexes, for every nice finite group G.

Background

For free G-spaces X and Y, the join defect measures the gap between the standard upper bound $$\operatorname{ind}_G(X*Y)\leq \operatorname{ind}_G X+\operatorname{ind}_G Y+1$$ and the actual equivariant index of the join. The paper proves that this defect is unbounded for every finite group that is neither cyclic of prime-power order nor generalized quaternion.

The open problem asks whether the same unboundedness holds for nice finite groups. The question is equivalently formulated as asking whether, for every integer n≥1, there are finite free G-simplicial complexes X and Y whose join defect is at least n.