I want to understand a simplified version of the general $k$-local structure theorem proved in the paper "Reductive group actions":
For $k$ a characteristic zero algebraically closed field, $G$ a connected reductive group over $k$, and for a homogeneous $G$-spherical variety $X=G/H$ for which $BH$ is open, let $P$ be the adapted parabolic and $P=LU$ its Levi decomposition, the local structure theorem states that we have an $P$-equivariant isomorphism $(L/L\cap H)\times U\cong BH$. This follows from the work of Brion-Luna-Vust and Knop.
For a general characteristic $0$ field $k$, Knop and Krotz also proved a generic $k$-local structure theorem: let $G$ be a connected reductive group over $k$, $X$ a $k$-dense $G$-spherical variety w.r.t the minimal parabolic subgroup $P$, let $Q_{k}(X)$ be the adapted parabolic of $X$ and $Q_{k}(X)=LU$ be its Levi decomposition, then the $k$-LST states that there exists a smooth affine $L$-subvariety $X_{el}\subseteq X$ such that the action of $L$ on $X_{el}$ is elementary and the natural morphism $U\times X_{el}\to X$ is an open embedding.
Suppose $G$ is a connected quasi-split group over $k$, and $X=G/H$ a homogeneous $k$-spherical variety, $B$ its Borel subgroup over $k$ such that $BH$ is open, $Q_{k}(X)$ its adapted parabolic with Levi decomposition $Q_{k}=LU$.
My question is: Is it true that similar to the previous setting over algebraically closed field, we have an $Q_{k}(X)$-equivariant isomorphism $(L/L\cap H)\times U\to BH$? that is to say, is $X_{el}$ a homogeneous $L$-variety when $X$ is homogeneous?
