$2$-distance list $(\mathrm{\Delta }+2)$-coloring of planar graphs with girth at least 10

Given a graph $G$ and a list assignment $L(v)$ for each vertex of $v$ of $G$. A proper $L$-list-coloring of $G$ is a function that maps every vertex to a color in $L(v)$ such that no pair of adjacent vertices have the same color. We say that a graph is list $k$-colorable when every vertex $v$ has a list of colors of size at least $k$. A $2$-distance coloring is a coloring where vertices at distance at most 2 cannot share the same color. We prove the existence of a $2$-distance list ($\mathrm{\Delta }+2$)-coloring for planar graphs with girth at least $10$ and maximum degree $\mathrm{\Delta }\ge 4$.