Shelling conjecture for the proposed order on $\max\nabla_{n,k}$

Let $\mathfrak{S}_{n,k}^r$ be the set of $r$-stable permutations, let $\Phi(\mathfrak{S}_{n,k}^r)$ be the associated set on which the graded colexicographic order is imposed, and let $\max\nabla_{n,k}$ be the corresponding set of maximal simplices. The construction in the paper gives a total order on $\mathfrak{S}_{n,k}^r$ and hence on $\max\nabla_{n,k}$. Shelling conjecture. The given order on $\max\nabla_{n,k}$ is a shelling order. This would extend the shelling order known in the $k=2$ case; the supplied text gives no proof or resolution for general $k$ and $r$.