Rainbow Hajnal--Szemerédi conjecture for graph systems

Let $t$ and $n$ be positive integers, and let $\textbf{G}=\{G_1,G_2,\ldots,G_{\frac{n}{t}\binom{t}{2}}\}$ be an $n$-vertex graph system. Rainbow Hajnal--Szemerédi conjecture. If

$$\delta(G_i)\geq\left(1-\frac{1}{t}\right)n$$

for every $i\in\left[\frac{n}{t}\binom{t}{2}\right]$, then $\textbf{G}$ admits a rainbow $K_t$-factor. This is the exact rainbow analogue of the Hajnal--Szemerédi theorem: the conjecture asks whether the sharp minimum-degree threshold remains sufficient when each clique edge must be selected from a distinct graph in the system.