Existence of unitary orthonormal bases for finite-index regular subfactors

Let $N\subset M$ be a finite-index regular inclusion of factors of type $II_1$. A unitary orthonormal basis conjecture asserts that $M/N$ has a unitary orthonormal basis.

The question arises from the known group-subfactor case, where such bases exist, and from an earlier asserted result whose proof appears incomplete because it relies on an argument for Cartan subalgebras. Whether every finite-index regular subfactor, without an irreducibility assumption, admits such a basis remains open.