Han–Zhao conjecture on vertex-disjoint cliques in multipartite graphs

Let $r\geq t\geq 3$ and $k\geq 2$, and let $K_{t}$ denote the complete graph on $t$ vertices. For positive integers $n_1,\ldots,n_r$, write $K_{n_1,\ldots,n_r}$ for the complete $r$-partite graph with part sizes $n_1,\ldots,n_r$, and let $\operatorname{ex}(n_1,\ldots,n_r,kK_t)$ be the maximum number of edges in an $r$-partite graph with these part sizes that contains no vertex-disjoint union of $k$ copies of $K_t$. Let $f(n_1,\ldots,n_r,k,t)$ be the function specified in the cited formulation. Assume that $n_r$ is sufficiently large. Han–Zhao's conjecture.

$$\operatorname{ex}(n_1,\ldots,n_r,kK_t)=f(n_1,\ldots,n_r,k,t).$$

This conjecture concerns the extremal number for forbidding $k$ vertex-disjoint copies of a clique in a multipartite graph. It extends the problem posed after earlier results for $kK_2$ and for general multipartite graphs, while the source does not state a resolution of the general case.