Normality and Cohen–Macaulayness of conormal Schubert varieties

Let $I$ be a $k$-subset of ${1,\ldots,n}$, and let $\mathit{CS}_I$ denote the conormal Schubert variety, equivalently the unique component of $\mu^{-1}(\mathcal O_{k,n})$ satisfying $p(\mathit{CS}_I)=S_I$. Normality and Cohen–Macaulayness conjecture. The variety $\mathit{CS}_I$ is Cohen--Macaulay and normal for all $I$. The analogous normality and rational-singularity properties are known for Schubert varieties, but the corresponding assertions for these conormal Schubert varieties are presented here as beliefs and remain unresolved.