Algebraic hyperbolicity of adjoint linear systems on spherical varieties

Moraga and Yeong conjectured that for a smooth complex projective variety $X$ of dimension $n$, an ample line bundle $A$ on $X$ and an integer $m \ge 3 n + 1$, very general elements of the adjoint linear system $|ω_{X} \otimes A^{\otimes m}|$ are algebraically hyperbolic. We prove the conjecture for spherical varieties with smooth orbit closures. As a corollary, we conclude that the conjecture holds for horospherical varieties, and for toroidal spherical varieties. Furthermore, for any spherical variety, we show that the conjecture holds modulo the complement of an open dense orbit.