Reduced tangent-cone conjecture for Richardson varieties

Let $X_u^v=X_u\cap X^v$ be a Richardson variety in $GL_n/B$, let $wB\in X_u^v$, and let the tangent cone at $wB$ be defined by the associated graded local ring of $X_u^v$ at that point. Richardson reduced-tangent-cone conjecture. The tangent cones of $X_u^v$ are reduced. The conjecture concerns the scheme-theoretic local geometry of Richardson varieties. The supplied text gives computational motivation involving Gröbner bases with squarefree monomial ideals, but does not establish reducedness in general.