The folk conjecture on von Neumann subalgebras of uniform Roe and quasi-local algebras

Let $X$ be an infinite uniformly locally finite metric space. A von Neumann algebra is said to embed into an associated uniform Roe algebra or quasi-local algebra if it admits an injective $*$-homomorphism into that algebra. For a countable collection of natural numbers $(n_k)_k$, write $\prod_k \mathrm{M}_{n_k}(\mathbb{C})$ for the corresponding von Neumann algebra.

Folk conjecture. The only von Neumann algebras that can embed into a uniform Roe algebra or a quasi-local algebra associated to a uniformly locally finite metric space are those of the form

$$\prod_k \mathrm{M}_{n_k}(\mathbb{C}),$$

where $(n_k)_k$ is a countable, possibly finite, bounded collection of natural numbers.

The conjecture concerns the possible von Neumann algebra subalgebras of uniform Roe and quasi-local algebras; the paper subsequently gives a more precise refinement. Its resolution is not specified in the supplied text.