2 problems

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.

Blackadar–Handelman Choquet simplex conjecture. The affine space $DF(A)$ is a Choquet simplex for any $C^*$-algebra $A$.