The model-completeness characterization for theories of tracial von Neumann algebras

Let $N$ be a tracial von Neumann algebra, and let $T$ be its theory. The algebra $N$ is of type I.

Model-completeness conjecture. The theory $T$ is model complete if and only if $N$ is of type I.

The preceding proposition gives a general model-theoretic criterion for model completeness under the Continuum Hypothesis, but the stated characterization for tracial von Neumann algebras is presented as a conjecture. Its resolution is not indicated in the supplied text.