Heckenberger–Vocke conjecture on the structure of graded right coideal subalgebras

Let $U_q(\mathfrak{g})$ be a quantum group, let $w tin W$, let $\phi$ be a character, and let $U^-[w]_{\phi}$ be the corresponding right coideal subalgebra. The associated graded algebra has a map

$$f:\operatorname{gr}(U^-[w]_{\phi})\to U^{\leq 0}$$

sending elements to their leading terms. Let $D$ be its image, let $M:=\langle K_{\mu}^{-1}\mid \mu\in\operatorname{supp}(\phi)\rangle$, and let $L$ be the quotient group of $M$.

Heckenberger–Vocke conjecture. The map $f$ is an injective homomorphism of $\mathbb{Z}$-graded right coideal subalgebras, and

$$D_0=M,\qquad DT_L=U^-[w']T_L,$$

where

$$w':=\left(\prod_{\beta\in\operatorname{supp}(\phi)}s_{\beta}\right)w, \qquad w=s_{\alpha_{k_1}}s_{\alpha_{k_2}}\cdots s_{\alpha_{k_m}}\in W.$$

Since the elements of $\operatorname{supp}(\phi)$ are pairwise orthogonal, $w'$ is obtained from $w$ by deleting all factors $s_{\alpha_i}$ for which $\beta_i\in\operatorname{supp}(\phi)$. The conjecture is proved for type $A_n$, but the general structure remains open.