Braun–Solus stable hypersimplex h-star unimodality conjecture

For positive integers $n$ and $r$, let $\Delta_{n,2}^{\operatorname{stab}(r)}$ denote the $r$-stable hypersimplex. Braun–Solus' conjecture. The $h^*$-vector of $\Delta_{n,2}^{\operatorname{stab}(r)}$ is unimodal for all $r$. The source records this as an open question; unimodality is known for $r=2$, $r=3$, and $r=\lfloor n/2\rfloor-1$.