Optimality of the round-robin policy for more than two sources

Let $M>2$ and let $N$ be the number of channels. For a problem instance $(M,N,\boldsymbol\mu)$, consider all possible permutations of arms in the set $\mathcal{S}(M)$ and the round-robin policy that schedules source $m\in[M]$ on channel $((m+t)\bmod M)+1$. Round-robin optimality conjecture. The round-robin policy is optimal for every problem instance $(M,N,\boldsymbol\mu)$. This conjecture generalizes the stated optimality theorem for $M=2$; the provided text gives no resolution, so its status remains open.