On Popa's factorial commutant embedding problem

An open question of Sorin Popa asks whether or not every $R^{\mathcal{U}}$-embeddable factor admits an embedding into $R^{\mathcal{U}}$ with factorial relative commutant. We show that there is a locally universal McDuff II$_1$ factor $M$ such that every property (T) factor admits an embedding into $M^{\mathcal{U}}$ with factorial relative commutant. We also discuss how our strategy could be used to settle Popa's question for property (T) factors if a certain open question in the model theory of operator algebras has a positive solution.