Graphs with girth $2\mathrm{\ell }$ and without longer even holes are $3$-colorable

For a number $\mathrm{\ell }\ge 2$, let ${\mathcal{H}}_{\mathrm{\ell }}$ denote the family of graphs which have girth $2\mathrm{\ell }$ and have no even hole with length greater than $2\mathrm{\ell }$. Wu, Xu, and Xu conjectured that every graph in ${\bigcup }_{\mathrm{\ell }\ge 2}{\mathcal{H}}_{\mathrm{\ell }}$ is 3-colorable. In this paper, we prove that every graph in ${\mathcal{H}}_{\mathrm{\ell }}$ is 3-colorable for any integer $\mathrm{\ell }\ge 5$.