Enumeration conjecture for non-Temperley–Lieb subfactor planar algebras at index $3+\sqrt{5}$

A subfactor planar algebra is non-Temperley–Lieb if it is not the Temperley–Lieb subfactor planar algebra. The index is the Jones index, and dual subfactor planar algebras are counted separately when listed as distinct examples. The notation $A_3*A_4$ denotes the Bisch–Jones Fuss–Catalan subfactor planar algebra; $2D2$, $3^{\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}}$, $3^{\mathbb{Z}/4\mathbb{Z}}$, and $4442$ name the specified subfactor planar algebras.

Enumeration conjecture. At index $3+\sqrt{5}$, there are exactly $13$ non-Temperley–Lieb subfactor planar algebras: the Bisch–Jones Fuss–Catalan $A_3*A_4$ subfactor planar algebra and its dual; the $5$ quotients of the Fuss–Catalan $A_3*A_4$ subfactor planar algebra, including the self-dual tensor product; the unique $2D2$ subfactor planar algebra and its dual; Izumi's unique symmetrically self-dual $3^{\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}}$ subfactor planar algebra; Izumi's unique $3^{\mathbb{Z}/4\mathbb{Z}}$ subfactor planar algebra and its dual; and the unique symmetrically self-dual $4442$ subfactor planar algebra.

This conjecture proposes completeness of the list at index $3+\sqrt{5}$, following constructions and classification results for several of the listed examples and for exactly $1$-supertransitive subfactor planar algebras. The supplied source does not state that the complete enumeration has been proved.