Vertex decomposability of higher independence complexes of trees

Let $T$ be a tree and let $r\geq 1$. The complex $\operatorname{Ind}_r(T)$ is the $r$-independence complex of $T$. Tree higher-independence conjecture. For any tree $T$ and $r\geq 1$, the complex $\operatorname{Ind}_r(T)$ is vertex decomposable. The conjecture extends the established result for caterpillar graphs. Computations verify vertex decomposability for all non-caterpillar trees with at most 10 vertices, but the general case remains open.