The homomorphism-threshold conjecture for simply connected graphs

Let $G$ be a simply connected graph and fix $r\geq 2$. Suppose $\phi:G\to H$ is a graph homomorphism such that $H$ is $C_{2r+1}$-free, and suppose further that some odd cycle $C$ of $G$ satisfies

$$|\phi(V(C))|\leq 2r+2.$$

Homomorphism-threshold conjecture. Then

$$\chi(G)\leq 2r.$$

This conjecture proposes an improved bound on the chromatic number in the setting of the paper's homomorphism-threshold theorem. Its status is not resolved in the supplied text.