Morrison–Peters principal graph conjecture for subfactors of index between 5 and $3+\sqrt{5}$

A subfactor has principal graphs, as an ordered pair, determined by its index in the range

$$5<[M:N]<3+\sqrt{5}.$$

Morrison–Peters conjecture. Any subfactor with index in the range $(5,3+\sqrt{5})$ has principal graphs $(A_\infty,A_\infty)$, the pair of graphs represented in the source by {\hspace{-3pt}\begin{array}{c}% \raisebox{-2.5pt}{\includegraphics[height=6mm]{diagrams/graphs/ \@nameuse{bwd1v1p1v1x1p0x1duals1v1x2}}}% \end{array}\hspace{-3pt}} on both sides, or the pair of graphs represented in the source by {\hspace{-3pt}\begin{array}{c}% \raisebox{-2.5pt}{\includegraphics[height=6mm]{diagrams/graphs/ \@nameuse{bwd1v1v1p1v1x0p0x1p0x1v0x1x0p1x0x1duals1v1v2x1x3}}}% \end{array}\hspace{-3pt}} on both sides. This conjecture concerns the classification of principal graphs for subfactors in the first range beyond index $5$; the paper reports that its results eliminate all remaining weeds with initial quadruple points up to index $3+\sqrt{5}$, providing evidence for the conjecture, but does not establish the full classification.