K_r-free correspondence-cover packing conjecture

Let $r\ge3$, let $\Delta$ be a maximum-degree bound, and let a correspondence $L$-packing have the meaning defined in the paper.

K_r-free correspondence-cover packing conjecture. For every $r\ge3$, there is some $C_r>0$ such that the following holds for

$$k=C_r\frac{\Delta}{\log\Delta}.$$

Suppose that $G$ and $H$ are graphs such that $H$ is a $k$-fold correspondence cover of $G$ via some $L$, $H$ contains no copy of $K_r$, and $\Delta(H)\le\Delta$. Then $H$ admits a correspondence $L$-packing.

This is proposed as a stronger speculation encompassing the triangle-free and bipartite cover-graph directions. The source gives no resolution.