Conjugacy conjecture for basic regular exact Borel subalgebras

Let $\Bbbk$ be an algebraically closed field, and let $A,B\subseteq R$ be basic regular exact Borel subalgebras.

Conjugacy conjecture. There exists an invertible element $x\in R$ such that

$$x^{-1}Ax=B.$$

This would strengthen uniqueness up to isomorphism for basic regular exact Borel subalgebras to uniqueness up to conjugation inside the same algebra, analogous to the Wedderburn--Malcev theorem for maximal semisimple subalgebras. The source presents this as an expected generalization, and no resolution is given.