Subfactorizability conjecture for three-dimensional fusion bialgebras of type II

Let $\b5$ be the parameter of the one-parameter family of three-dimensional canonical fusion bialgebras in case II, with $\b5\rfloor 3$. A fusion bialgebra is subfactorizable if it can be realized as the $2$-box space of a planar algebra arising from a subfactor. Subfactorizability conjecture. The one-parameter family of three-dimensional fusion bialgebras in case II can be subfactorized if and only if

$$\b5=\cot^2\left(\frac{\pi}{2N+2}\right)$$

for some $N\in\mathbb{Z}_+$. The preceding realization results show that the corresponding planar algebras arise from subfactors for precisely these parameter values; the conjecture asserts the same characterization for subfactorizability of the fusion bialgebras themselves.