Stronger subcubic strong edge-coloring conjectures

Let $G$ be a subcubic graph. Stronger subcubic conjectures. (a) If $G$ is bridgeless and is neither the Wagner graph nor the graph formed from $K_{3,3}$ by subdividing one edge, then $\chi^s(G)\le9$. (b) If $G$ is bridgeless and $|V(G)|\ge13$, then $\chi^s(G)\le8$. (c) If $G$ has girth at least $5$, then $\chi^s(G)\le7$. Parts (a) and (b) are attributed to results of HLL and LMSS in the source, while part (c) strengthens the still-open bipartite no-4-cycle assertion; the supplied status evidence therefore leaves this combined candidate open.