The conjecture that Paley type II Hadamard matrix subfactors have Temperley–Lieb standard invariants

Let $H$ be a Paley type $II$ Hadamard matrix, and let the associated spin model subfactor be the subfactor generated by $H$. Its standard invariant is the collection of higher relative commutants and associated planar-algebraic data of the subfactor.

Paley type $II$ Temperley–Lieb conjecture. Paley type $II$ Hadamard matrix subfactors have Temperley–Lieb standard invariants.

The authors motivate this conjecture by numerical computations and by the known behavior of Paley type $I$ examples. They also state that Paley type $II$ Hadamard matrix subfactors are infinite depth; whether their standard invariants are Temperley–Lieb remains open.