Geometric properties conjecture for degenerate flag varieties on boundary faces

Let $d53c$ be the parameter defining the degeneration, let $F^{\mathbf{c}}$ be the corresponding face, and let $\mathcal{F}^{\mathbf{e}}(\lambda)$ be the $\mathbf{e}$-degenerate flag variety associated to a dominant weight $\lambda$, where

$$\mathbf{e}\in F^{\mathbf{c}}\setminus\mathrm{relint}(F^{\mathbf{c}}).$$

Geometric properties conjecture. For every such $\mathbf{e}$, $\mathcal{F}^{\mathbf{e}}(\lambda)$ is normal, Cohen–Macaulay, and has rational singularities. For parameters in $\mathrm{relint}(F^{\mathbf{c}})$, these properties follow because the degenerate flag variety is isomorphic to a Schubert variety. Whether the boundary degenerations are themselves Schubert varieties is unknown, and the conjecture asserts that they nevertheless retain these geometric properties.