Classification conjecture for ultraproducts of matrix algebras

Let $(n_i)_{i\to\theta}$ and $(m_j)_{j\to\theta'}$ be families of natural numbers, and let $\theta$ and $\theta'$ be ultrafilters. Write $\tau$ for the corresponding ultralimit of residues modulo $d$. Ultraproduct classification conjecture.

$$\prod_\mathcal{U} M_{n_i} \equiv \prod_{\operatorname{\mathcal{V}}} M_{m_j} \qquad \text{if and only if} \qquad \lim_\mathcal{U} n_i \bmod d = \lim_{\mathcal{V}} m_i \bmod d \quad \text{for all } d \in \mathbb{N}.$$

This proposes that the equivalence of ultraproducts of finite-dimensional matrix algebras is completely determined by the limiting residue classes of their matrix sizes modulo every natural number. The surrounding discussion concerns structural decompositions of pseudocompact and pseudomatrical $C^*$-algebras; the supplied text does not establish the conjecture or indicate whether it has been resolved.