Furstenberg discretization with moment control and entropy rigidity

Let $G$ be a locally compact group, let $Γ$ be a lattice in $G$, and let $η$ be a spread-out probability measure on $G$. A Furstenberg discretization of $η$ to $Γ$ is a probability measure $μ$ on $Γ$ such that every $(G,η)$-stationary space is also $(Γ,μ)$-stationary. For cocompact lattices $Γ$ and