Classification conjecture for Borel subalgebras of quantum groups

Let $C=U^-[w]_{\phi^-}T_L\psi(U^+[w^+])_{\phi^+}$ be a triangular right coideal subalgebra, with $\operatorname{supp}(\phi^+)=\operatorname{supp}(\phi^-)$ and $L=\operatorname{supp}(\phi^+)^{\perp}$. For every homogeneous element $x_\beta$ of degree $\beta$, define

$$\psi(x_\beta)=q^{-(\beta,\beta)/2}x_\beta K_\beta^{-1}.$$

Let $w'$ be the Weyl-group element associated with $w$ and the support of the character as in the preceding conjecture.

Classification conjecture. The algebra $C$ is a Borel subalgebra if and only if

$$w^+w'^{-1}=w_0.$$

This gives a proposed classification of triangular Borel subalgebras satisfying the stated support and orthogonality conditions; its general validity is not established in the source.