A graph $G$ is said to satisfy the Vizing bound if $\chi(G)\le \omega(G)+1$, where $\chi(G)$ and $\omega(G)$ denote the chromatic number and clique number of $G$, respectively. The class of graphs satisfying the Vizing bound is clearly $\chi$-bounded in the sense of Gyarfas. It has been conjectured that if $G$ is triangle-free and fork-free, where the fork is obtained from $K_{1,4}$ by subdividing two edges, then $G$ satisfies the Vizing bound. We show that this is true if, in addition, $G$ is $C_5$-free.