Törnquist's universality conjecture for isomorphism of factors

A von Neumann algebra is separably acting when it acts on a separable Hilbert space. For equivalence relations $E$ and $F$ on standard Borel spaces, $E$ is Borel reducible to $F$ if there is a Borel function sending $E$-equivalent points to $F$-equivalent points and non-equivalent points to non-equivalent points. An orbit equivalence relation induced by a continuous action of the unitary group on a Polish space is the relation identifying points in the same orbit under a continuous action of a unitary group on a Polish space.

Törnquist's conjecture. The isomorphism relation for separably acting factors is universal, with respect to Borel reducibility, for orbit equivalence relations induced by a continuous action of the unitary group on a Polish space. In fact, this universality already holds for isomorphism of $\operatorname{II}_1$ factors.

The conjecture asks whether the known upper bound on the complexity of isomorphism for von Neumann algebras is optimal. The source states that it had been made publicly in talks; no resolution is supplied here.