Transversal Hajnal–Szemerédi conjecture

Let $k\geq 2$ be an integer, and let $n$ be a sufficiently large multiple of $k$. Let

$$\bm{G}=\left(G_1,G_2,\ldots,G_{\frac{n}{k}\binom{k}{2}}\right)$$

be a collection of graphs on a common vertex set of size $n$. Write $\delta(\bm{G})$ for the minimum degree over all graphs in the collection, and let a transversal copy of a $K_k$-factor mean a collection of vertex-disjoint copies of $K_k$ covering all vertices, using exactly one edge from each graph in $\bm{G}$. Transversal Hajnal–Szemerédi conjecture. If

$$\delta(\bm{G})\geq\left(1-\frac{1}{k}\right)n,$$

then $\bm{G}$ contains a transversal copy of a $K_k$-factor.

This is presented as a transversal analogue of the Hajnal–Szemerédi theorem and would generalise that theorem. The source gives no resolution, so the conjecture remains open.