Equivalence of abelian regular centralizers and Borel parabolic conjecture

Let $X=G/H$ be an affine homogeneous spherical variety without type $N$ roots, and let $P(X)$ be the associated parabolic subgroup. Abelian-centralizer equivalence conjecture. The following conditions are equivalent: (1) $X$ has abelian regular centralizers; (2) $P(X)$ is a Borel subgroup. The paper states this as an expected equivalence and provides no general proof.