Affine spherical homogeneous spaces with good quotient by a maximal unipotent subgroup

For an affine spherical homogeneous space $G/H$ of a connected semisimple algebraic group $G$, we consider the factorization morphism by the action on $G/H$ of a maximal unipotent subgroup of $G$. We prove that this morphism is equidimensional if and only if the weight semigroup of $G/H$ satisfies a simple condition.
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