Nice-group topological Hedetniemi conjecture for free simplicial complexes

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 $\operatorname{ind}\mathcal{K}$ be the least $k$ admitting a $G$-equivariant map to $E_kG$. Equip Cartesian products with the diagonal $G$-action. Nice-group topological Hedetniemi conjecture. For every pair $\mathcal{K}_1,\mathcal{K}_2$ of finite free $G$-simplicial complexes,

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

The paper presents this as its weakest and most general remaining conjectural form: the theorem rules out the corresponding assertion for groups that are not nice, while the nice-group case remains open.