Ferroni–Matherne–Vecchi's real-rootedness conjecture for characteristic Chow polynomials

Let $P$ be a finite graded bounded poset, and let $P$ be Cohen–Macaulay. Denote by the characteristic Chow polynomial of $P$ the polynomial associated with $P$ in the Chow-polynomial construction. Ferroni–Matherne–Vecchi's conjecture. The characteristic Chow polynomial of $P$ is real-rooted. This extends the matroid Chow-polynomial real-rootedness conjecture to Cohen–Macaulay posets; the source does not give a resolution.