Brändén–Leander gamma-positivity conjecture for lecture hall order polytopes

Let $P$ be a naturally labeled, graded poset with nonnegative rank function $\rho$, and set ${\boldsymbol s}=\rho+1$. Let $O(P,{\boldsymbol s})$ denote the corresponding ${\boldsymbol s}$-lecture hall order polytope, and let $h^*(O(P,{\boldsymbol s});z)$ be its Ehrhart $h^*$-polynomial. Brändén–Leander's gamma-positivity conjecture. The polynomial $h^*(O(P,{\boldsymbol s});z)$ is $\gamma$-nonnegative. The source presents this as one of two conjectures for lecture hall order polytopes and records it as open in the general setting.