Flexibility conjecture for affine spherical varieties

Let $G$ be a reductive group and let $X$ be an affine spherical $G$-variety, meaning that $X$ is an irreducible normal algebraic $G$-variety on which a Borel subgroup of $G$ has an open orbit. Suppose that $X$ has no non-constant invertible regular functions, equivalently ${\mathbb K}[X]^{\times}={\mathbb K}^{\times}$.

Flexibility conjecture. $X$ is flexible.

This conjecture concerns the relationship between spherical group actions and flexibility, a strong form of transitivity generated by additive-group actions. It was proposed in the cited work; the supplied source does not indicate whether it has been resolved.