Multipartite Hajnal–Szemerédi perfect packing conjecture

Let $k\ge 2$ and let $G$ be a $k$-partite graph with parts $V_1,\ldots,V_k$ of the same size $n$. Define the partite minimum degree of $G$ to be the largest integer $d$ such that every vertex has at least $d$ neighbors in each part other than its own. A perfect $K_k$-packing is a collection of vertex-disjoint $k$-cliques covering all vertices of $G$. Fischer–Kühn–Osthus's conjecture. If the partite minimum degree of $G$ is at least $\left(1-\frac{1}{k}\right)n$, then $G$ has a perfect $K_k$-packing unless both $k$ and $n$ are odd and $G$ is isomorphic to a single exceptional graph. The statement is presented as an application conjecture in the source; its resolution status is not given.