The refined embedding conjecture for von Neumann algebras in uniform Roe and quasi-local algebras

Let $X$ be a uniformly locally finite metric space. For a countable, possibly finite, bounded collection of natural numbers $(n_k)_k$, consider the von Neumann algebra $\prod_k\mathrm{M}_{n_k}(\mathbb{C})$. A collection $(X_k)_k$ is a disjoint collection of uniformly bounded subsets when the subsets are pairwise disjoint and their diameters are uniformly bounded.

Refined embedding conjecture. The only von Neumann algebras that can embed into

$$\mathrm{C}^*_u(X) \quad\text{and}\quad \mathrm{C}^*_{\operatorname{ql}}(X)$$

are those of the form $\prod_k\mathrm{M}_{n_k}(\mathbb{C})$. Moreover, $\prod_k\mathrm{M}_{n_k}(\mathbb{C})$ embeds into $\mathrm{C}^*_u(X)$ or $\mathrm{C}^*_{\operatorname{ql}}(X)$ if and only if $X$ contains a disjoint collection $(X_k)_k$ of uniformly bounded subsets satisfying

$$|X_k|=n_k$$

for all $k$.

The conjecture refines the preceding folk conjecture by characterizing exactly when the matrix-product von Neumann algebras embed. The supplied text establishes embeddings from suitable disjoint uniformly bounded subsets, but gives no resolution of the converse or of the full classification.