Nice-group conjecture for cross-indices of face posets

Let $G$ be a nice group, meaning a cyclic $p$-group or a generalized quaternion group whose order is a power of $2$. For a finite free $G$-simplicial complex $\mathcal{K}$, let $\mathcal{F}(\mathcal{K})$ be its face poset, and let $\operatorname{x-ind}$ denote the cross-index used in the source. Nice-group face-poset conjecture. For every pair $\mathcal{K}_1,\mathcal{K}_2$ of finite free $G$-simplicial complexes,

$$\operatorname{x-ind}\bigl(\mathcal{F}(\mathcal{K}_1)\times\mathcal{F}(\mathcal{K}_2)\bigr)=\min\{\operatorname{x-ind}\mathcal{F}(\mathcal{K}_1),\operatorname{x-ind}\mathcal{F}(\mathcal{K}_2)\}.$$

This is presented as a weaker open form of the preceding poset conjecture and concerns the cross-index after passing to face posets.