Hibi–Tsuchiya–Olsen regular unimodular triangulation conjecture for lecture hall simplices

Let ${\boldsymbol s}\in\mathbb{Z}_{\geq 1}^n$ be given, and let $\mathbf{P}_n^{({\boldsymbol s})}$ denote the corresponding ${\boldsymbol s}$-lecture hall simplex. Hibi–Tsuchiya–Olsen's triangulation conjecture. The simplex $\mathbf{P}_n^{({\boldsymbol s})}$ has a regular, unimodular triangulation. The source says this is known only in particular cases and that no counterexample is known in the remaining cases; it is also important for results concerning the ${\boldsymbol s}$-canonical triangulation of ${\boldsymbol s}$-lecture hall order polytopes.