The Margulis-type regular subalgebra conjecture for II$_1$ factors

Let $M$ be a separable II$_1$ factor, and let $P\subset M$ be a regular von Neumann subalgebra, meaning that the normalizer of $P$ in $M$ generates $M$. Margulis-type conjecture. There exists a separable II$_1$ factor $M$ such that every regular von Neumann subalgebra $P\subset M$ is either finite dimensional or of finite index.

The conjecture is presented as a von Neumann algebraic analogue of Margulis' normal subgroup theorem. The preceding corollary gives evidence for it by ruling out certain crossed-product decompositions, but the methods do not exclude crossed products with infinite-dimensional $2$-cocycles.