The classification conjecture for irreducible subfactor planar algebras of index between 5 and $3+\sqrt{5}$

Let a subfactor planar algebra mean the planar algebra associated with a finite-index subfactor, and let $A_\infty$ denote the infinite-depth principal graph. Consider irreducible subfactor planar algebras whose index lies in the interval $(5,3+\sqrt{5})$. Classification conjecture. The only two irreducible subfactor planar algebras, apart from those with principal graph $A_\infty$, are the two quantum group subfactors described above, arising from the 3-dimensional representations of $SU(2)$ and $SU(3)$ at a 14th root of unity. Computer evidence suggests that there are no other finite-depth subfactors in this interval, but a complete classification remains open; the two stated examples are known to be unique for their principal graphs.