Z2-indices and Hedetniemi's conjecture

The $\mathbb{Z}_2$-index ${\rm ind}(X)$ of a $\mathbb{Z}_2$-CW-complex $X$ is the smallest number $n$ such that there is a $\mathbb{Z}_2$-map from $X$ to $S^n$. Here we consider $S^n$ as a $\mathbb{Z}_2$-space by the antipodal map. Hedetniemi's conjecture is a long standing conjecture in graph theory concerning the graph coloring problem of tensor products of finite graphs. We show that if Hedetniemi's conjecture is true, then ${\rm ind}(X \times Y) = \min \{ {\rm ind}(X) , {\rm ind}(Y)\}$ for every pair $X$ and $Y$ of finite $\mathbb{Z}_2$-complexes.