Setup

Let $G$ be a (connected) reductive group over an algebraically closed field $k$, and fix a Borel subgroup $B \subset G$ and a maximal torus $T \subset B$. Let $\lambda: \mathbb{G}_m \to T$ be a dominant one-parameter subgroup. For any complete normal $G$-variety $X$ (by "variety" I mean separated, integral, and finite-type over $k$), it is a general fact that the Białynicki-Birula decomposition of $X$ with respect to $\lambda$ has a big cell. In other words, there exists a connected component $S \subset X^{\mathbb{G}_m}$ such that $\lim_{t \to 0} \lambda(t)x \in S$ for all $x$ in some open subset of $X$. Moreover, $S$ is a normal variety and carries an action of the reductive group $M_\lambda = \{g \in G\ |\ \lambda(t)g = g\lambda(t)\ \forall t \in \mathbb{G}_m\}$.

I am interested in the case where $\mathrm{char}(k) = 0$ and $X$ is a smooth projective spherical $G$-variety. Then, $S$ is a spherical $M_\lambda$-variety, and I believe $S$ is also smooth and projective (because the fixed point scheme $X^{\mathbb{G}_m}$ is closed, and Iversen has proven that $X^{\mathbb{G}_m}$ is smooth). When $X$ is in fact a toroidal spherical variety, $S$ is also toroidal, and a nice paper by Knop shows that all the combinatorial data of the spherical variety $S$ (i.e. its spherical homogeneous datum and its colored fan) are completely determined by the combinatorial data of $X$.

My Question: what can be said about the combinatorial data of $S$ when $X$ is smooth and projective but not necessarily toroidal?

A Few Ideas

It seems likely to me that Knop's results for toroidal varieties should at least partly hold in this setting. For one thing, it is well-known that there exists a $G$-equivariant birational morphism $\pi: X' \to X$ with $X'$ complete and toroidal. Knop's results hold for $X'$, and it seems to me that some of these results should pass down to $X$. I am also thinking that the local structure theorem along with Luna's étale slice theorem may be enough to repeat at least some of Knop's arguments for the toroidal case. Indeed, some of Knop's arguments use the local structure theorem on an open subset $U \subset X$ to get $U \cong R_u(P) \times Z$, where $P = \{g \in G\ |\ gU = U\}$ and $Z$ is an affine spherical variety under the action of a Levi subgroup $M \subset P$. In the toroidal case, $Z$ is in fact toric for a quotient of $M$, which is the key to Knop's argument. In the smooth but not necessarily toroidal case, $Z$ is smooth affine and spherical, so Luna's étale slice theorem gives $Z \cong M \times^H V$, where $H$ is reductive, $M/H$ is an affine spherical $M$-variety and $V$ is a spherical $H$-module (see Corollary 2.2 of this paper). So, it seems like the structure of $Z$ might still be nice enough to relate $\lim_{t \to 0} \lambda(t)z$ for $z \in Z$ to the combinatorial data of $Z$ and $X$.

I intend to try to work out these approaches in more detail myself, but I'm not particularly familiar with these sorts of arguments, so I would really appreciate any help, intuition, or related results that anyone is willing to offer!