Beck et al.'s triangulation conjecture for the lecture hall cone

Let $\mathscr{R}_n$ be the lecture hall cone in the special case associated with $(1,2,\ldots,n)$, let $\mathfrak{S}_{n-1}$ be the symmetric group, and let $\mathrm{des}(\pi)$ denote the number of descents of $\pi$. Beck et al.'s triangulation conjecture. There exists a regular, flag, unimodular triangulation of $\mathscr{R}_n$ admitting a shelling order such that its maximal simplices are indexed by $\pi\in\mathfrak{S}_{n-1}$, with each simplex attached along $\mathrm{des}(\pi)$ of its facets. This would provide additional geometric information about the lecture hall cone; the source records the question as open.