  The strong chromatic index of a graph $G$, denoted $\chi_s'(G)$, is the least
number of colors needed to edge-color $G$ so that edges at distance at most two
receive distinct colors. The strong list chromatic index, denoted
$\chi_{s,\ell}'(G)$, is the least integer $k$ such that if arbitrary lists of
size $k$ are assigned to each edge then $G$ can be edge-colored from those
lists where edges at distance at most two receive distinct colors. We use the
discharging method, the Combinatorial Nullstellensatz, and computation to show
that if $G$ is a subcubic planar graph with $\operatorname{girth}(G) \geq 41$
then $\chi_{s,\ell}'(G) \leq 5$, answering a question of Borodin and Ivanova
[Precise upper bound for the strong edge chromatic number of sparse planar
graphs, Discuss. Math. Graph Theory, 33(4), (2014) 759--770]. We further show
that if $G$ is a subcubic planar graph and $\operatorname{girth}(G) \geq 30$,
then $\chi_s'(G) \leq 5$, improving a bound from the same paper. Finally, if
$G$ is a planar graph with maximum degree at most four and
$\operatorname{girth}(G) \geq 28$, then $\chi_s'(G) \leq 7$, improving a more
general bound of Wang and Zhao from [Odd graphs and its application on the
strong edge coloring, arXiv:1412.8358] in this case.
