Conjecture A on graded character-shifted quantum Schubert cells

Let $W$ be the Weyl group, let $w\in W$, and let $\phi$ be a character. Give $U^-_q(\mathfrak{g})$ the $\mathbb{Z}$-grading determined by $\deg(E_{\alpha_i})=1$, $\deg(K_{\alpha_i})=0$, and $\deg(F_{\alpha_i})=-1$. Let $f:\operatorname{gr}(U^-[w]_{\phi})\to U^{\leq 0}$ be the canonical map sending an element to its summands of leading degree, and write $D=\operatorname{im}(f)$. Let $D^0$ be the degree-zero part, set $G(D^0)=\\{K_{\mu}^{-1}\mid \mu\in\operatorname{supp}(\phi)\\}$, and let $\widetilde{G}$ denote its quotient group or localization. Define $w'=\left(\prod_{\beta\in\operatorname{supp}(\phi)}s_{\beta}\right)w$. Conjecture A. The algebra $D$ is a $\mathbb{Z}$-graded right coideal subalgebra of $U^{\leq 0}$, its degree-zero part is the semigroup algebra $D^0=\Bbbk[G(D^0)]$, and

$$\Bbbk[\widetilde{G}(D^0)]D=\Bbbk[\widetilde{G}(D^0)]U^-[w'].$$

Morally, this identifies the graded character shift with $U^-[w']$ up to localization and gives an explicit formula for the Weyl group element governing the graded algebra. The source presents this as an important structural conjecture for understanding the representation theory of right coideal subalgebras; its resolution status is not established in the supplied text.