Flexibility of affine spherical varieties

We prove that the automorphism group $\mathrm{A}\mathrm{u}\mathrm{t}(X)$ of an affine spherical variety $X$ acts transitively on the set of smooth points $X^{reg}\mathrm{.}$ If every invertible regular function on $X$ is constant, we prove that $X$ is flexible, i.e., the subgroup of $\mathrm{A}\mathrm{u}\mathrm{t}(X)$ generated by all ${\mathbb{G}}_a$-subgroups acts transitively on $X^{reg}\mathrm{.}$