Li's conjecture on generalized connectivity of complete multipartite graphs

Let $G$ be a complete multipartite graph, let $k$ be a positive integer, and let $S$ be a $k$-subset of $V(G)$. Li's conjecture. We have $\kappa_k(G)=\kappa(S)$ when the numbers of vertices in the parts of $G$ that are not in $S$ are almost the same. The conjecture generalizes the preceding complete equipartite formulation; the supplied text gives no resolution status.