Interlacing conjecture for Chow polynomials of simplicial posets

Let $P$ be a simplicial poset, with associated polynomials $\operatorname{H}_{\widehat{P}}(x)$, $\operatorname{H}_{{\widehat{P}}^\ast}(x)$, and augmented polynomial $\operatorname{H}^{\operatorname{aug}}_{\widehat{P}}(x)$. Interlacing conjecture. The polynomials

$$\operatorname{H}_{\widehat{P}}(x),\quad \operatorname{H}_{{\widehat{P}}^\ast}(x),\quad \operatorname{H}^{\operatorname{aug}}_{\widehat{P}}(x)$$

are real-rooted. Moreover, the roots of both $\operatorname{H}_{\widehat{P}}(x)$ and $\operatorname{H}_{{\widehat{P}}^\ast}(x)$ interlace the roots of

$$\operatorname{H}^{\operatorname{aug}}_{\widehat{P}}(x)=\operatorname{H}^{\operatorname{aug}}_{{\widehat{P}}^\ast}(x).$$

The claim concerns real-rootedness and interlacing for Chow polynomials associated with simplicial posets; the supplied text gives no resolution beyond noting that related interlacing conjectures had appeared previously.