Casselgren–Petrosyan conjecture for outerplanar graphs

Let $G$ be an outerplanar graph, and let $\mu_{\operatorname{int}}(G)$ denote its interval coloring impropriety. Casselgren–Petrosyan's outerplanar graph conjecture.

$$\mu_{\operatorname{int}}(G) \leq 2.$$

This would improve the known bound $\mu_{\operatorname{int}}(G)\leq \left\lceil \Delta(G)/4\right\rceil+1$. The source states that the conjecture has been proved when $\Delta(G)\leq 8$, but leaves the general case unresolved.