A spectral strong approximation theorem for measure-preserving actions

Let $\unicode[STIX]{x1D6E4}$ be a finitely generated group acting by probability measure-preserving maps on the standard Borel space $(X,\unicode[STIX]{x1D707})$. We show that if $H\leq \unicode[STIX]{x1D6E4}$ is a subgroup with relative spectral radius greater than the global spectral radius of the action, then $H$ acts with finitely many ergodic components and spectral gap on $(X,\unicode[STIX]{x1D707})$. This answers a question of Shalom who proved this for normal subgroups.