The tensor-product conjecture for minimal intermediate subfactors

Let $N_i\subset M_i$, $i=1,2$, be irreducible subfactors with finite index, and let $n_i$ be the dimension of the second higher relative commutant of $N_i\subset M_i$. Consider the tensor-product inclusion

$$N_1\otimes N_2\subset M_1\otimes M_2.$$

A minimal intermediate subfactor is one minimal above the lower endpoint. The intermediate subfactors of the form $N_1\otimes P$ or $P\otimes N_2$ are excluded.

Tensor-product conjecture. The number of minimal intermediate subfactors in $N_1\otimes N_2\subset M_1\otimes M_2$ not of either excluded form is at most

$$(n_1-1)(n_2-1).$$

This conjecture is motivated by a lemma bounding maximal subgroups of direct products and by the corresponding direct-product result for Wall's conjecture. The supplied text does not state a general proof.