The strengthened cycle-free chromatic discrepancy conjecture

Let $G$ be a graph. For an integer $\ell\ge 3$, call $G$ $C_{\ell+1}$-free if it contains no cycle of length exactly $\ell+1$ as a, not necessarily induced, subgraph. Let $K_{\ell}$ denote the complete graph on $\ell$ vertices, and let $\varphi(G)$ denote the chromatic discrepancy of $G$.

Strengthened cycle-free chromatic discrepancy conjecture. For every integer $\ell\ge 3$, every $C_{\ell+1}$-free graph $G\ne K_{\ell}$ satisfies

$$\varphi(G)\ge \chi(G)-\ell+1.$$

The source presents this as a stronger conjecture motivated by the suspicion that equality in the weaker bound occurs only when $\ell=2$ or when $G$ is complete. Its status is not otherwise resolved in the supplied text.