The reduced tangent-cone conjecture for K-orbit closures

Let $p\in Y_{\gamma}$ be a point of the $K$-orbit closure, and let $\operatorname{gr}_{\mathfrak m_p}\mathcal O_{p,Y_{\gamma}}$ be the associated graded local ring. Reduced tangent-cone conjecture.

$$\operatorname{Spec}(\operatorname{gr}_{\mathfrak m_p}\mathcal O_{p,Y_{\gamma}})$$

is reduced. This was checked exhaustively for all cases with $p+q\leq 7$, but remains open in general.