Outerplanar graph star chromatic index conjecture

Let $G$ be an outerplanar graph with maximum degree $\Delta\ge 4$, and let $\chi'_{\mathrm{star}}(G)$ denote its star chromatic index, the smallest number of colors in a proper edge coloring with no bichromatic path or cycle of length four.

Outerplanar graph star chromatic index conjecture. Every such graph satisfies

$$\chi'_{\mathrm{star}}(G)\le \left\lfloor\frac{3\Delta}{2}\right\rfloor+1.$$

Known bounds give $\chi'_{\mathrm{star}}(G)\le \left\lfloor\frac{3\Delta}{2}\right\rfloor+12$, with the additive constant reducible to $9$ by more involved analysis. The conjecture remains open.