Correspondence packing number for triangle-free graphs

Let $G$ be a graph and let $\chi^\star_c(G)$ denote its correspondence packing number. Let $\Delta$ be a maximum-degree bound.

Triangle-free correspondence packing conjecture.

$$\chi^\star_c(G) \le (1+o(1)) \frac{\Delta}{\log \Delta}$$

for any triangle-free graph $G$ with $\Delta(G)\le\Delta$, as $\Delta\to\infty$.

The paper proves this asymptotic bound for bipartite graphs, and notes that the complete bipartite examples show the bipartite result is sharp up to an asymptotic factor of $2$. Extending it to all triangle-free graphs is left open.