Gallai's conjecture for chordal graphs with subdivided caterpillar representations

Let $G$ be a connected chordal graph admitting a tree representation $T$, where $T$ is a subdivided caterpillar. Subdivided-caterpillar Gallai conjecture. Then

$$\mathrm{lpt}(G)=1.$$

This asserts that every such graph has a single-vertex longest path transversal. It is a restricted version of the broader open problem asking whether every connected chordal graph satisfies $\mathrm{lpt}(G)=1$; the supplied text does not state whether this restricted claim is resolved.