Commutative coherent configurations suffice for matrix multiplication exponent two

A coherent configuration is a finite set equipped with a partition of its ordered pairs into basis relations; it is commutative when its adjacency algebra is commutative. A coherent configuration realizes the matrix multiplication tensor $\langle n,n,n\rangle$ when the associated tensor realization has matrix dimensions $n,n,n$, and its rank is the number of basis relations.

Commutative coherent configuration conjecture. There exist commutative coherent configurations ${\cal C}_n$ realizing $\langle n,n,n\rangle$ and of rank $n^{2+o(1)}$.

Such configurations would provide a route to proving that the matrix multiplication exponent satisfies $\omega=2$. The paper gives commutative coherent configurations yielding nontrivial upper bounds on $\omega$, but the asserted rank-$n^{2+o(1)}$ family is not established here.