  We say that a graph $G$ has an {\em odd $K_4$-subdivision} if some subgraph
of $G$ is isomorphic to a $K_4$-subdivision and whose faces are all odd holes
of $G$. For a number $\ell\geq 2$, let $\mathcal{G}_{\ell}$ denote the family
of graphs which have girth $2\ell+1$ and have no odd hole with length greater
than $2\ell+1$. Wu, Xu and Xu conjectured that every graph in
$\bigcup_{\ell\geq2}\mathcal{G}_{\ell}$ is 3-colorable. Recently, Chudnovsky et
al. and Wu et al., respectively, proved that every graph in $\mathcal{G}_2$ and
$\mathcal{G}_3$ is 3-colorable. In this paper, we prove that no
$4$-vertex-critical graph in $\bigcup_{\ell\geq5}\mathcal{G}_{\ell}$ has an odd
$K_4$-subdivision. Using this result, Chen proved that all graphs in
$\bigcup_{\ell\geq5}\mathcal{G}_{\ell}$ are 3-colorable.
