Structural matching conjecture for dense random graphs

Fix $r\geqslant 4$, and let $G\sim G(n,q)$, where

$$n^{-2/(r-1)}(\log n)^{1/{r-1 \choose 2}}\ll q=q(n)\ll n^{-2/r}.$$

For each $r\in\mathbb{N}$, let $S_r(G)$ be the set of vertices in a largest $K_r$-matching of $G$, with an arbitrary deterministic choice in case of ties.

Structural matching conjecture. With high probability as $n\rightarrow\infty$, the graph $G-S_r(G)$ contains a $K_{r-1}$-matching that covers all but $o(\sqrt{\mu_r})$ vertices of $V(G)\setminus S_r(G)$.

This conjecture would provide the structural theorem needed to extend the paper's results to larger values of $r$. The source notes that the analogous case $r=3$ follows from an earlier structural result, while the cases $r\geqslant4$ remain open.