Nice-group conjecture for the cross-index of products of free simplicial posets

Let $G$ be a nice group, meaning a cyclic $p$-group or a generalized quaternion group whose order is a power of $2$. Let $\mathcal{P}_1$ and $\mathcal{P}_2$ be finite free $G$-simplicial posets, and let $\operatorname{x-ind}$ denote the cross-index used in the source. Nice-group cross-index conjecture.

$$\operatorname{x-ind}(\mathcal{P}_1\times\mathcal{P}_2)=\min\{\operatorname{x-ind}\mathcal{P}_1,\operatorname{x-ind}\mathcal{P}_2\}.$$

The authors present this as their strongest open conjecture; their main theorem shows that the analogous assertion cannot hold for arbitrary finite groups.