Conjecture that every extremal graph is complete multipartite

Let $\bm{k}=(k_1,\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt},k_s)$ be a non-increasing sequence of natural numbers, and let a graph be $\bm{k}$-extremal if it attains the maximum number $F(n;\bm{k})$ of $\bm{k}$-valid edge-colourings among graphs on $n$ vertices. A graph is complete multipartite if its vertex set can be partitioned into parts such that every pair of vertices in distinct parts is adjacent. Extremal multipartite conjecture. For every $\bm{k}$, every $\bm{k}$-extremal graph is complete multipartite. At least one extremal graph is known to be complete multipartite, but the conjecture asserts this for every extremal graph.