Spherical tropicalization and Berkovich analytification

Let $X$ be a spherical variety. We show that Tevelev and Vogiannou's tropicalization map from $X$ to its tropicalization factors through the Berkovich analytification $X^{\text{an}}$, as in the case for toric varieties. Furthermore we show that the tropicalization is a strong deformation retraction of $X^{\text{an}}$. We also give a strong deformation retraction of Thuillier's analytification $X^{\beth}$ onto a subspace described using the colored fan of $X$.