Conjecture B on triangular Borel subalgebras of quantum groups

Let $C$ be a triangular right coideal subalgebra of the form

$$C=U^-[w_-]_{\phi_-}\Bbbk[L]S(U^+[w_+])_{\phi_+}.$$

Here $w_+,w_-\in W$, $\phi_+$ and $\phi_-$ are characters, $L$ is the associated lattice, $S$ is the antipode, and $w_-'$ denotes the Weyl group element obtained from $w_-$ by the transformation prescribed in Conjecture A. Conjecture B. The algebra $C$ is a Borel subalgebra if and only if

$$w_+w_-'^{-1}=w_0, \qquad \ell(w_+)+\ell(w_-'^{-1})=\ell(w_0),$$

where $w_0$ is the longest Weyl group element. This proposed criterion aims to determine exactly which triangular right coideal subalgebras are Borel subalgebras. The source gives no resolution evidence beyond noting that Conjecture A and its corollary have been proved in many cases, so the status remains open in the supplied text.