4 problems

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

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

Chromatic discrepancy conjecture. Every integer $s\ge 2$ and every locally $s$-colourable graph $G$ satisfy

Chromatic-discrepancy conjecture. The chromatic discrepancy of $G$ is at least $\chi-\omega$.