Ardila–Billey conjecture on spread-out simplices

Let $\Delta_{d-1}$ be the standard $(d-1)$-simplex, and let $n\Delta_{d-1}$ denote its dilation by $n$. A unit simplex is a simplex of normalized volume one in $n\Delta_{d-1}$, and a collection of such simplices is spread out when it satisfies the spread-out condition of Ardila and Billey. Spread-out simplices conjecture. A collection of $n$ unit simplices in $n\Delta_{d-1}$ can be extended to a tiling if and only if it is spread out. This characterizes exactly which collections of unit simplices occur as part of a tiling; the supplied text gives no resolution status.