Regular-centralizer determination of the spherical root system

Let $X=G/H$ be an affine homogeneous spherical variety with abelian regular centralizers and no type $N$ roots. Let $J_A$ be the associated group scheme and let $J_A^{\mathrm{Knop}}$ be Knop's group scheme. Regular-centralizer determination conjecture.

$$J_A=J_A^{\mathrm{Knop}}.$$

In particular, the spherical root system of $X$ should be encoded entirely by its regular centralizers. The source gives no proof or resolution status for this assertion in the stated generality.