Conjecture on the complexity of theories of infinitely generic operator algebras

An infinitely generic operator algebra is the corresponding generic model in the sense used in the paper. For a structure $M$, let $\operatorname{Th}(M)$ denote its first-order theory; second-order arithmetic is the standard second-order arithmetic theory, and $A$ Turing reduces to $B$ when $A\leq_T B$.

Infinitely generic theories conjecture. Second-order arithmetic Turing reduces to both theories

$$\operatorname{Th}(M_{\mathrm{C}^*}),\qquad \operatorname{Th}(M_{\mathrm{tr}}),$$

of infinitely generic $\mathrm{C}^*$-algebras and infinitely generic tracial von Neumann algebras, respectively.

The analogous reduction is known for infinitely generic groups, yielding continuum many theories of existentially closed groups. The operator-algebraic assertion is presented as a conjecture; the source notes that, if it also held for embeddable tracial von Neumann algebras, it would have consequences for the theory of the hyperfinite II$_1$ factor.