Barker's tensor-product characterization of Choquet simplices

Let $K_1$ and $K_2$ be compact convex sets. Write $K_1 \otimes_* K_2$ and $K_1 \otimes^* K_2$ for their minimal and maximal tensor products of compact convex sets, respectively.

Barker's conjecture. $K_1 \otimes_* K_2 = K_1 \otimes^* K_2$ if and only if one of $K_1$ and $K_2$ is a Choquet simplex.

This would replace the square in the Namioka–Phelps characterization with an arbitrary compact convex set that is not a Choquet simplex, and is stated as an open problem first formulated by G. P. Barker.