Bóna–Dimitrov's generalized Sturm sequence 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 $UUD$. A sequence of real-rooted polynomials is a generalized Sturm sequence when consecutive terms satisfy $F_i(x)\preccurlyeq F_{i+1}(x)$, where $\preccurlyeq$ denotes the interlacing relation. Bóna–Dimitrov's generalized Sturm sequence conjecture. For any fixed $k\geq 1$, the polynomial sequence $\{W_{n,k}(x)\}_{n\geq k}$ is a generalized Sturm sequence. The real-rootedness of these polynomials is known, but this interlacing conjecture was proposed by Bóna et al. and its resolution is not specified here.