Okounkov–Bezrukavnikov filtration conjecture for rational Cherednik category O

Let $H_m$ be the rational Cherednik algebra, let $\mathcal{O}_m$ be its category $\mathcal{O}$, let $M_m(\lambda)$ be a Verma module, and let $s^m_\lambda$ be the stable-basis vector for the Hilbert scheme with parameter $m$. Okounkov–Bezrukavnikov filtration conjecture. Every representation in $\mathcal{O}_m$ admits a filtration compatible with the filtration on $H_m$, whose associated graded Verma modules have bigraded Frobenius character $s^m_\lambda$; the filtration is compatible with induction, restriction, and morphisms; the categorical action admits a filtered lift agreeing with the $U_q\widehat{\mathfrak{gl}}_b$ action, with $q$ recording filtration shift; and the filtration on finite-dimensional simples $L_m$ agrees with the filtration constructed by Gorsky, Oblomkov, Rasmussen, and Shende. This conjecture connects categorical actions on rational Cherednik category $\mathcal{O}$ with stable bases and is explicitly presented as open in the paper.