Conjecture on arithmetic complexity of theories of existentially closed II$_1$ factors

An embeddable tracial von Neumann algebra is one that embeds into an ultrapower of the hyperfinite II$_1$ factor $\mathcal R$. An existentially closed (e.c.) II$_1$ factor is an existentially closed model in the relevant theory, and $A\leq_T B$ means that $A$ Turing reduces to $B$.

Arithmetic-complexity conjecture. First-order arithmetic Turing reduces to the theory of any (embeddable) e.c. II$_1$ factor.

The source recalls the analogous result for any e.c. group. It presents the II$_1$-factor statement as a conjecture, and relates the embeddable case to the theory of the hyperfinite II$_1$ factor.