The cycle-free chromatic discrepancy conjecture

Let $G$ be a graph. For an integer $\ell\ge 2$, call $G$ $C_{\ell+1}$-free if it contains no cycle of length exactly $\ell+1$ as a, not necessarily induced, subgraph. Let $\varphi(G)$ denote the chromatic discrepancy of $G$.

Cycle-free chromatic discrepancy conjecture. For every integer $\ell\ge 2$, every $C_{\ell+1}$-free graph $G$ satisfies

$$\varphi(G)\ge \chi(G)-\ell.$$

This is proposed as a weaker version of the locally $s$-colourable conjecture, because every $C_{\ell+1}$-free graph is locally $\ell$-colourable. The source explicitly states that this weaker form is open.