Bóna–Dimitrov's Sturm-unimodality conjecture for Dyck-path polynomials

Let $W_{n,k}(x)=\sum_{m=0}^{k}w_{n,k,m}x^m$, where $w_{n,k,m}$ counts Dyck paths of semilength $n$ with $k$ occurrences of $UD$ and $m$ occurrences of $UUD$. For real-rooted polynomials, $G(x)\preccurlyeq F(x)$ denotes the interlacing relation, and a finite sequence is Sturm-unimodal if it increases under this relation up to some index and then decreases under the reverse relation. Bóna–Dimitrov's Sturm-unimodality conjecture. For any fixed $n\geq 1$, the polynomial sequence $\{W_{n,k}(x)\}_{1\leq k\leq n}$ is Sturm-unimdoal. The underlying polynomials are known to be real-rooted; the proposed Sturm-unimodality assertion remains unresolved in the supplied material.