The Cohen–Macaulay and Hilbert-polynomial conjectures for K-orbit closures

Let $p\in Y_{\gamma}$, let $\mathfrak m_p$ be the maximal ideal of the local ring $\mathcal O_{p,Y_{\gamma}}$, and define the $H$-polynomial by

$$\operatorname{Hilb}(\operatorname{gr}_{\mathfrak m_p}\mathcal O_{p,Y_{\gamma}},q)=\frac{H_{p,Y_{\gamma}}(q)}{(1-q)^{\dim Y_{\gamma}}}.$$

Hilbert-polynomial conjecture. The following assertions hold: (i) $\operatorname{gr}_{\mathfrak m_p}\mathcal O_{p,Y_{\gamma}}$ is Cohen–Macaulay; (ii) $H_{p,Y_{\gamma}}(q)\in\mathbb Z_{\geq 0}[q]$; and (iii) $H_{p,Y_{\gamma}}(q)\in\mathbb Z_{\geq 0}[q]$ is upper-semicontinuous. These properties are motivated by the analogous properties of Kazhdan–Lusztig–Vogan polynomials; the general assertions remain open.