Symmetric product-state conjecture for maximal overlap of permutation-invariant states

Let a permutation-invariant multipartite state be given, and consider its maximal overlap with product states. The question concerns whether the product state attaining this maximal overlap can be chosen with identical single-party factors. Symmetric product-state conjecture. Is it true that, in order to obtain the maximal overlap of any permutation-invariant state, we can assume the product state to be a tensor product of the same single-party state? The question asks whether optimization over arbitrary product states can always be reduced to optimization over symmetric product states. The supplied context says that only a partial answer follows, so the general claim is not established here.