Polynomial-growth conjecture for minimal intermediate subfactors

Let $N\subseteq M$ be an irreducible subfactor with finite index, and let the number of minimal intermediate subfactors be counted over the inclusion $N\subseteq M$. Polynomial-growth conjecture. There are constants $c_1,c_2$ such that, for every such inclusion, the number of minimal intermediate subfactors is less than

$$c_2[M:N]^{c_1}.$$

The paper proves an exponential upper bound for the number of minimal intermediate subfactors and conjectures that polynomial growth should hold instead. The source gives no resolution of this conjecture.