The algebra-object decomposition conjecture for the _1-_3 subfactor

Let $N\subset M$ be the $\mathscr{I}_1$-$\mathscr{I}_3$ subfactor constructed above, and let $\gamma$ be the corresponding algebra object in $\mathscr{I}_1$. Here $\nu_j$ and $\mu_j$ denote the simple objects indexed by $j$, as in the fusion-category notation of the paper.

Algebra-object decomposition conjecture. The algebra object satisfies

$$\gamma\cong 1+\sum_{j=1}^{\frac{n-1}{2}}(\nu_j+\mu_j).$$

This conjecture identifies the algebra object associated with the constructed subfactor and would determine the corresponding principal-graph data. The paper states that it is verified in the cases $n=3$ and $n=5$, but cannot be verified for arbitrary $n$.