Conjecture on the tensor structure of type $II_{\infty}$ von Neumann algebras in quantum gravity

Let $\mathcal{A}$ be a type $II_{\infty}$ von Neumann algebra with a tensor-product decomposition

$$\mathcal{A}\cong\mathcal{B}(\mathcal{H}_I)\overline{\otimes}P_0\mathcal{A}P_0.$$

Here $\mathcal{H}_I$ is an ordinary Hilbert space, $\mathcal{B}(\mathcal{H}_I)$ is its full algebra of bounded operators, and $P_0\mathcal{A}P_0$ is the internal type $II_1$ algebra associated with a finite projector $P_0$.

Quantum-gravity tensor-structure conjecture. The tensor-product structure above describes the first steps away from the semiclassical approximation toward a full theory of quantum gravity: $\mathcal{H}_I$ and $\mathcal{B}(\mathcal{H}_I)$ describe ordinary quantum matter interacting locally around each macroscopic point with the underlying gravitational degrees of freedom encoded in $P_i\mathcal{A}P_i$ or $P_0\mathcal{A}P_0$.

The conjecture proposes an interpretation of the type $II_{\infty}$ decomposition in which ordinary quantum matter is represented by the exterior Hilbert-space algebra and gravitational degrees of freedom by the internal type $II_1$ factor. It is a physical proposal about the transition beyond semiclassical quantum field theory, and the source gives no evidence that it has been proved or refuted.