Non-realizability of $A_{\aleph_0\aleph_1}$ as a graph CCR algebra

Let $\mathcal Z$ be the Jiang--Su algebra and let

$$A_{\aleph_0\aleph_1}:=M_{2^\infty}\otimes\bigotimes_{\aleph_1}\mathcal Z.$$

For a graph $G$, let $B(G)$ denote the associated CCR algebra. Non-realizability claim. The algebra $A_{\aleph_0\aleph_1}$ is not isomorphic to $B(G)$ for any graph $G$.

The algebra $A_{\aleph_0\aleph_1}$ is a direct limit of full matrix algebras but is not UHF, whereas every algebra $B(G)$ is AF and every separable algebra of this form is UHF. The source does not provide evidence resolving whether this specific non-realizability claim is open.