Li's conjecture on generalized connectivity of complete equipartite graphs

Let $G$ be a complete equipartite $a$-partite graph with partition $(X_1,X_2,\ldots,X_a)$, and let $k=ab+c$, where $b,c$ are integers and $0\leq c\leq a-1$. For a $k$-subset $S$ of $V(G)$ satisfying

$$|S\cap X_1|=\cdots=|S\cap X_c|=b+1$$

and

$$|S\cap X_{c+1}|=\cdots=|S\cap X_a|=b,$$

Li's conjecture. We have $\kappa_k(G)=\kappa(S)$. This is one of two conjectures proposed by W. Li in her Ph.D. thesis; the supplied text gives no resolution status.