Wrochna's index conjecture for products of $\mathbb{Z}_2$-spaces

Let $X$ and $Y$ be $\mathbb{Z}_2$-spaces, with product equipped with the diagonal $\mathbb{Z}_2$-action, and let $\operatorname{ind}(X)$ denote the index of a $\mathbb{Z}_2$-space. Wrochna's index conjecture. For all $\mathbb{Z}_2$-spaces $X$ and $Y$,

$$\operatorname{ind}(X\times Y)=\min\{\operatorname{ind}(X),\operatorname{ind}(Y)\}.$$

The conjecture remains open in general; the source notes that it holds if one factor is a sphere and that the cases of spheres of dimensions $0$ and $1$ follow from multiplicativity of $K_2$ and $K_3$.