Virtual matrix distributions classify nonstandard ergodic filtrations

Let $c4$ and $c4'$ be finitely isomorphic ergodic filtrations. Assume that both filtrations are nonstandard and that both have virtual matrix distributions for the sequences of associated semimetric triples. Virtual matrix distribution conjecture. If these virtual matrix distributions coincide, then the filtrations are isomorphic. This would provide an invariant for classifying nonstandard filtrations whose metric triples do not converge to an ordinary metric triple; whether it is complete in the stated setting remains open.