Conjecture on $\mathscr{C}^{N,1,1}$ and Xu's conformal-inclusion category

Let $N$ be a positive integer, and let $\mathscr{C}^{N,1,1}$ be the unitary fusion category constructed from the Yang--Baxter relation planar algebra. Let Xu's bimodule category be the unitary fusion category associated with the conformal inclusion

$$SU(N+2)_{N}\subset SU\left(\frac{(N+2)(N+1)}{2}\right)_{1}.$$

The $\mathscr{C}^{N,1,1}$ conjecture. The unitary fusion category $\mathscr{C}^{N,1,1}$ is isomorphic to Xu's bimodule category for this conformal inclusion. The source verifies the corresponding branching formula in the case $N=3$ and states that the branching formula is new for $N\geq4$; the general categorical identification remains open.