Approximate bicommutant conjecture for separable factors

Let $\mathcal{M}$ be a separable factor, and let $\mathcal{A}\subseteq\mathcal{M}$ be a separable unital $C^*$-subalgebra. The relative approximate bicommutant of $\mathcal{A}$ in $\mathcal{M}$ is the corresponding asymptotic bicommutant taken relative to $\mathcal{M}$. Approximate bicommutant conjecture. Every separable unital $C^*$-subalgebra of $\mathcal{M}$ equals its relative approximate bicommutant in $\mathcal{M}$. This generalizes Hadwin's asymptotic bicommutant theorem for $\mathcal{B}(\mathcal{H})$ and the theorem established in separable type $\mathrm{III}$ factors; the type $\mathrm{II}$ case is presented as requiring new methods and remains open.