The horospherical spherical amoeba convergence conjecture

Let $G/H$ be a spherical homogeneous space, let $Y\subseteq G/H$ be a subvariety, and let $\operatorname{sLog}_{\Gamma,t}(Y)$ be its spherical amoeba and $\operatorname{strop}(Y)$ its spherical tropicalization. Spherical amoeba convergence conjecture. If $G/H$ is horospherical, then the spherical amoebae $\operatorname{sLog}_{\Gamma,t}(Y)$ approach $\operatorname{strop}(Y)$ in the sense of Kuratowski convergence as $t\to0$. This is presented as a conjectural positive answer to the spherical amoeba convergence question; the corresponding assertion for general spherical homogeneous spaces is expected to be more difficult and may require non-Archimedean geometry.