The reduced-product rigidity conjecture for II$_1$-factors

Assume $\mathrm{OCA}_\infty+\mathrm{MA}_{\aleph_1}$. Let $(M_i)$ and $(N_i)$ be sequences of separably representable type II$_1$ factors, and let $\bigoplus_\tau M_i$ be the ideal of sequences whose normalized traces tend to zero, with analogous notation for $(N_i)$. Reduced-product rigidity conjecture for II$_1$-factors.

$$\prod M_i/\bigoplus_\tau M_i\cong\prod N_i/\bigoplus_\tau N_i$$

if and only if there are finite sets $F_0,F_1\subseteq\mathbb N$ and a bijection

$$g\colon\mathbb N\setminus F_0\to\mathbb N\setminus F_1$$

such that $M_i\cong N_{g(i)}$. The conjecture proposes that, under these forcing axioms, an isomorphism between reduced products is determined by matching all but finitely many factor algebras. The source presents it as an analogy with its preceding rigidity results and gives no resolution here.