The classification conjecture for subfactors of index between 5 and $3+\sqrt{5}$

Let $N\subset M$ be a subfactor with index in the interval $(5,3+\sqrt{5})$. Its principal graph is the graph describing the associated standard invariant; write $A_\infty$ for the infinite $A$-type principal graph. The paper also specifies two further principal-graph pairs, denoted here by $\Gamma(\mathcal A)$ and $\Gamma(\mathcal B)$.

Classification conjecture. Any subfactor with index in $(5,3+\sqrt{5})$ has principal graph $A_\infty$, $\Gamma(\mathcal A)$, or $\Gamma(\mathcal B)$.

The theorem preceding this statement proves the assertion for 1-supertransitive subfactors, while the conjecture removes that hypothesis and is left open by the paper.