Countable-dimensionality conjecture for locally finite Steinitz numbers

A Steinitz number $\tau=\prod_{p\in\mathbb{P}}p^{r_p}$ is locally finite if $r_p<\infty$ for every prime $p\in\mathbb{P}$. Let $A$ be a unital locally matrix algebra, and let $\mathbf{n}(A)$ denote its Steinitz number.

Countable-dimensionality conjecture. If $\tau$ is a locally finite Steinitz number, $A$ is a unital locally matrix algebra, and $\mathbf{n}(A)=\tau$, then $A$ is countable dimensional.

This concerns the dimension of unital locally matrix algebras whose Steinitz number has only finite prime exponents. The supplied text does not state whether the claim is known or resolved.