The lecture-hall Sperner triangulation conjecture

Let $R_n$ be the lecture-hall cone, and let $S_{n-1}$ denote the symmetric group on $n-1$ letters. For a permutation $\pi\in S_{n-1}$, write $des(\pi)$ for its number of descents. A Sperner $2$-pair of $[n-1]$ is a pair of subsets $A,B\subset [n-1]$ such that neither $A$ nor $B$ is contained in the other.

Lecture-hall Sperner conjecture. There exists a regular, flag, unimodular triangulation of $R_n$ that admits a shelling order such that the maximal simplices of the triangulation are indexed by $\pi\in S_{n-1}$ and each such simplex is attached along $des(\pi)$ of its facets. Further, the number of minimal non-edges in the triangulation is the number of Sperner $2$-pairs of $[n-1]$.

The conjecture is motivated by the analogous braid triangulation of the cube $[0,1]^{n-1}$ and experimental evidence for $n\leq 7$. The authors state that it has proved to be a real challenge, and a corresponding triangulation remains elusive.