Unimodality conjecture for $r$-stable second hypersimplices

Let $\Delta_{n,2}^{\operatorname{stab}(r)}$ denote the $r$-stable second hypersimplex, and let its Ehrhart $h^\ast$-polynomial be the polynomial whose coefficient sequence is the $h^\ast$-vector. Unimodality conjecture. The $h^\ast$-polynomials of the $r$-stable second hypersimplices $\Delta_{n,2}^{\operatorname{stab}(r)}$ are unimodal. The preceding computations establish this in several cases, including $r=2$ and $r=3$ under the stated hypotheses, but the general assertion remains open.