Let $G$ be a semisimple group over $\mathbb{C}$ and let $X=G/H$ be a spherical homogeneous space,
then $X$ defines a spherical datum (Luna datum) $\mathcal L(X)=(N,\mathcal V, \mathcal D, \rho,\varsigma)$, see below.

The spherical datum is a functor. Indeed, assume we have two spherical homogeneous spaces $X_1$ and $X_2$ of $G$.
A $G$-morphism $\varphi\colon X_1\to X_2$ induces a morphism $\mathcal L(\varphi)\colon \mathcal L(X_1)\to \mathcal L(X_2)$ (in the obvious sense, see below).

Losev's Theorem 1 from his paper [Uniqueness property for spherical homogeneous spaces](http://arxiv.org/abs/0904.2937) says that if there exists an *isomorphism* $\lambda\colon\mathcal L(X_1)\to \mathcal L(X_2)$, then there exists an isomorphism
$\varphi\colon X_1\to X_2$.

> **Question.** Assume  we have two spherical homogeneous spaces $X_1$ and $X_2$ of $G$, and a *morphism* of spherical data 
$\lambda\colon \mathcal L(X_1)\to \mathcal L(X_2)$.
Does there exist a $G$-morphism $\varphi\colon X_1\to X_2$  *inducing* $\lambda$, that is, such that $\lambda=\mathcal L(\varphi)$?

A positive answer to this question would strengthen Losev's theorem even in the case when $\lambda$ is an isomorphism.

We specify our version of the spherical datum of $X$.
Let $B\subset G$ be a Borel subgroup, and let $T\subset B\subset G$ be a maximal torus.
Let $S=S(G,T,B)$ denote the corresponding system of simple roots.
Let ${\mathcal{P}}(S)$ denote the set of subsets of $S$.

Let ${\mathcal{X}}(B)$ denote the character group of $B$, and let $M\subset {\mathcal{X}}(B)$ denote the weight lattice of $X$.
Set $N={\rm Hom}(M,{\mathbb{Z}})$, $N_{\mathbb{Q}}=N\otimes_{\mathbb Z}{\mathbb{Q}}$.
Let $\mathcal{V}\subseteq N_{\mathbb{Q}}$ denote the valuation cone of $X$.

Let ${\mathcal{D}}$ denote the set of colors of $X$ (the set of $B$-invariant prime divisors of $X$). We have two maps:
$$\rho\colon {\mathcal{D}}\to N,\qquad \varsigma\colon {\mathcal{D}}\to {\mathcal{P}}(S). $$
Here, for $D\in{\mathcal{D}}$,
$$ \varsigma(D)=\{\alpha\in S\ | \ P_\alpha\cdot D\neq D\}, $$
where $P_\alpha\supset B$ denotes the parabolic subgroup corresponding to $\alpha\in S$.
We set $\mathcal L(X)=(N,\mathcal V, \mathcal D, \rho,\varsigma)$.

If $\mathcal L(X_i)=(N_i,\mathcal V_i,\mathcal D_i, \rho_i,\varsigma_i)$ for $i=1,2$,
then a $G$-morphism $\varphi\colon X_1\to X_2$ induces a morphism of spherical data
$$ \mathcal L(\varphi)=(\ \lambda_N\colon N_1\to N_2,\quad \lambda_{\mathcal D}\colon\mathcal D_1\to\mathcal D_2\ ) $$
such that $(\lambda_N\otimes_{\mathbb Z} \mathbb Q)(\mathcal V_1)\subseteq \mathcal V_2$ and the corresponding  diagrams for $\varsigma_i$ and for $\rho_i$ commute: for any $D_1\in\mathcal D_1$ we have
$$\lambda_N(\rho_1(D_1))=\rho_2(\lambda_{\mathcal D}(D_1)),\qquad
      \varsigma_1(D_1)=\varsigma_2(\lambda_{\mathcal D}(D_1)). $$
This permits us to define an abstract morphism of spherical data $\lambda\colon \mathcal L(X_1)\to\mathcal L(X_2)$ and to ask our question above.