On invariant subalgebras of noncommutative Poisson boundaries for higher rank lattices

Let $G$ be a real connected semisimple Lie group with trivial center, no non-trivial compact factors, and all simple factors of real rank at least two.

Let $\Gamma<G$ be an irreducible lattice, let $P<G$ be a minimal parabolic subgroup, and consider the crossed product $L^\infty(G/P,\nu_P)\rtimes \Gamma$.

We prove that every $\Gamma$-invariant von Neumann subalgebra of $L^\infty(G/P,\nu_P)\rtimes \Gamma$ is of the form $L^\infty(G/Q,\nu_Q)\rtimes \Lambda$, where $P\leq Q\leq G$ and $\Lambda\lhd\Gamma$.

This confirms a conjecture of Amrutam--Hartman.