Impropriety bound for 2-trees

Let $G$ be a 2-tree, that is, a graph obtained from a triangle by repeatedly adding a new vertex adjacent to both endpoints of an existing edge. Let $\mu_{\operatorname{int}}(G)$ denote the impropriety of an interval coloring of $G$. 2-tree impropriety conjecture.

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

The conjecture is motivated by the fact that the smallest 2-tree, $K_3$, has impropriety $2$, while the source reports no 2-tree with larger impropriety. Its status is unresolved in the supplied text.