2.  If $G$ is planar and has girth at least $41$, then $\chi_s'(G) \leq 5$.

\[thm:wz\] Fix $d \geq 4$ and let $G$ be a graph with $\Delta(G) \leq d$.

1.  If $G$ has girth at least $2d-1$ and $\operatorname{mad}(G) < 2 + \frac{2}{6d-7}$, then $\chi_s'(G) \leq 2d-1$.

2.  If $G$ is planar and has girth at least $10d-4$, then $\chi_s'(G) \leq 2d-1$.

One barrier to proving sparsity conditions that imply ${\chi_{s}'(G)}\leq 5$ is that there exist graphs $G$ with $\operatorname{mad}(G) = 2$ and ${\chi_{s}'(G)}= 6$. Let $S_3$ be a triangle with pendant edges at each vertex, and let $S_4$ be a $4$-cycle with pendant edges at two adjacent vertices. For $k \geq 5$, let $S_k$ be a $k$-cycle with pendant edges at each vertex. Each of $S_3$, $S_4$ and $S_7$ have maximum average degree $2$ and strong chromatic index at least 6, see Figure \[fig:S3S4S5\]. However, these graphs are 6-critical with respect to ${\chi_{s}'(G)}$, as the removal of any edge from $S_3$, $S_4$ or $S_7$ results in a graph that has a strong edge-coloring using five colors.

Our main theorem demonstrates that if these few graphs are avoided, and the maximum average degree is not too large, then we 