Sturm-unimodality conjecture for refined Narayana polynomials

Let $W_{n,k}(t)$ be the polynomials considered in the paper. A finite sequence of real-rooted polynomials is Sturm-unimodal if it increases by interlacing up to one index and then decreases by reverse interlacing. Sturm-unimodality conjecture. For any fixed $n\geq 1$, the sequence $\{W_{n,k}(t)\}_{1\leq k\leq n}$ is Sturm-unimodal. This is a stronger interlacing property than coefficient or value unimodality, and remains conjectural in the source.