Strict outerness versus trivial bicentralizer for amenable group actions

Establish whether every faithful trace-preserving action of an amenable locally compact group on a II_1 factor has trivial bicentralizer if and only if it is strictly outer, namely whether $(M,\alpha)=\mathbb C1$ and faithfulness are equivalent to $M'\cap(M\rtimes_\alpha G)=\mathbb C1$.

Background

The paper defines the bicentralizer $(M,\alpha)$ of a continuous trace-preserving action $\alpha:G\curvearrowright M$ on a tracial von Neumann algebra by asymptotic commutation with bounded nets that are uniformly almost invariant on compact subsets of $G$. Triviality of this algebra is proposed as an analogue of the classical bicentralizer condition for type III factors.

For amenable groups, strict outerness is expressed by the triviality of the relative commutant in the crossed product, $M'\cap(M\rtimes_\alpha G)=\mathbb C1$. The paper proves substantial special cases, including discrete groups and groups with a compact open subgroup, but leaves the general connected, noncompact case unresolved.