Multiplicativity of higher-dimensional spheres

Prove or disprove that every sphere S^d with d ≥ 2 is multiplicative as a Z_2-space; equivalently, determine whether ind(X × Y) = min{ind(X), ind(Y)} for all Z_2-spaces X and Y.

Background

The conjecture of Wrochna asserts that the Z_2-index is multiplicative under products of Z_2-spaces. This would imply that every antipodal sphere is multiplicative. The cases S0 and S1 are known, while the cases of dimensions at least two remain unresolved.

References

On the other hand, it remains open whether $Sd$ is multiplicative for all $d \ge 2$.

— A survey on Hedetniemi's conjecture  (2502.16078 - Zhu, 22 Feb 2025) in Section 2.4, paragraph following Theorem 2.?? on multiplicative Z_2-spaces