A strong edge-coloring of a graph $G$ is an edge-coloring such that any two edges on a path of length three receive distinct colors. We denote the strong chromatic index by $chi_{s}(G)$ which is the minimum number of colors that allow a strong edge-coloring of $G$. ErdH{o}s and Nev{s}etv{r}il conjectured in 1985 that the upper bound of $chi_{s}(G)$ is $frac{5}{4}Delta^{2}$ when $Delta$ is even and $frac{1}{4}(5Delta^{2}-2Delta +1)$ when $Delta$ is odd, where $Delta$ is the maximum degree of $G$. The conjecture is proved right when $Deltaleq3$. The best known upper bound for $Delta=4$ is 22 due to Cranston previously. In this paper we extend the result of Cranston to list strong edge-coloring, that is to say, we prove that when $Delta=4$ the upper bound of list strong chromatic index is 22.
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