Is any spherical subgroup conjugate to a subgroup defined over a smaller algebraically closed field?

Question

Let $G_0$ be a connected semisimple algebraic group defined over an algebraically closed field $k_0$. Let $k\supset k_0$ be a larger algebraically closed field. We write $G=G_0\times_{k_0} k$ for the base change of $G_0$ from $k_0$ to $k$.

Question. Let $H\subset G$ be a spherical subgroup of $G$ (defined over $k$). Is it true that $H$ is always conjugate to a (spherical) subgroup defined over $k_0$?

In other words, is it true that there exists an element $g\in G(k)$ and a (spherical) subgroup $H_0\subset G_0$ (defined over $k_0$) such that $$ gHg^{-1}=H_0\times_{k_0} k\ ?$$

Answer 1

The morphism $G/H\to\textrm{Spec}\,k$ is defined and flat over over some finitely generated $k_0$-subalgebra of $k$. Thus, the question amounts to whether a flat family of homomogeneous spherical $k_0$-varieties is generically trivial in the étale toplogy. In characteristic zero, this is true by Thm. 3.1 of Alexeev, V.; Brion, M.: Moduli of affine schemes with reductive group action. J. Algebraic Geom. 14 (2005), no. 1, 83–117. In positive characteristic, the problem is completely open, as far as I know.