Zaslavsky's multipartite graph conjecture for maximum frustration index

Let $K_{n_1,n_2,\ldots,n_k}$ be a complete multipartite graph with $k\geq 3$. Let $-K_{n_1,n_2,\ldots,n_k}$ denote the signature in which every edge is negative, and write $l(-G)$ and $l_{\max}(G)$ for the negative and maximum frustration indices of a graph $G$. Zaslavsky's multipartite graph conjecture. Complete multipartite graphs $K_{n_1,n_2,\ldots,n_k}$ with $k\geq 3$ attain maximum frustration index when signed all negative, that is,

$$l(-K_{n_1,n_2,\ldots,n_k})=l_{\max}(K_{n_1,n_2,\ldots,n_k}).$$

The conjecture is refuted by the chordal graph $K_{3,1,1}$, which is also a complete multipartite graph with $k=3$, since $l(-K_{3,1,1})=1<2=l_{\max}(K_{3,1,1})$.