Best-bound conjecture for the star chromatic index of outerplanar graphs

Let $G$ be an outerplanar graph with maximum degree $\Delta$. The outerplanar star chromatic index conjecture.

$$\chi_{\mathrm{star}}'(G) \leq \lfloor 1.5\Delta \rfloor + 1.$$

This would improve the known bound $\chi_{\mathrm{star}}'(G)\leq \lfloor 1.5\Delta\rfloor+5$ for outerplanar graphs and is presented as the conjectured best upper bound for the star chromatic index. Its resolution is not specified in the supplied text.