Győri's asymptotic clique-packing conjecture

Let $G$ be an $n$-vertex graph. For $r\text{ fixed}$, write $u_r(G)$ for the maximum number of edge-disjoint $r$-cliques in $G$, and let $t_{r-1}(n)$ denote the number of edges in the Turán graph with $r-1$ parts. If $k\in\mathbb{R}$ satisfies

$$e(G)=t_{r-1}(n)+k,$$

Győri's conjecture. For every fixed $r\geq 3$,

$$\nu_r(G)\geq \frac{(2-o(1))k}{r}.$$

This conjecture asymptotically strengthens the Győri–Tuza bound on decomposing the edges of a graph into $2$-cliques and $r$-cliques, and extends the expected statement to $r=3$.