Regular unimodular triangulation conjecture for lecture hall polytopes

Let $P$ be a naturally labeled ranked poset, let $\rho$ be its rank function, and set $s=\rho+1$. The associated lecture hall polytope is $O(P,s)$. Regular unimodular triangulation conjecture. The polytope $O(P,s)$, or some related polytope with the same Ehrhart polynomial, has a regular and unimodular triangulation. If true, this would imply unimodality of the coefficients of $A_{(P,s)}(t)$ through the relationship between Ehrhart $h^*$-polynomials and regular unimodular triangulations.