Robust Hajnal–Szemerédi conjecture for induced $K_r$-factors

Let $r\geq 2$, let $n$ be divisible by $r$, and let $G$ be an $(\tfrac{(r-1)n}{r}+1)$-regular graph on $n$ vertices. A subset of $V(G)$ induces a $K_r$-factor if its induced subgraph has a partition into vertex-disjoint copies of $K_r$. Robust Hajnal–Szemerédi conjecture. For every $r\geq 2$, there is an $\varepsilon>0$ such that at least

$$\varepsilon 2^n$$

subsets of $V(G)$ induce a $K_r$-factor.

This is a robust analogue of the Hajnal–Szemerédi theorem, replacing the existence of one $K_r$-factor by a positive proportion of subsets inducing one. The supplied source gives no resolution status.