Given a finite index inclusion, $N\subset M$, of $II_1$ factors we can construct two towers of finite dimensional algebras known as the $\textit{standard invariant}$. For low index, this has allowed for a complete calculation of subfactors. However, at index 6 this breaks downs because there is an uncountable family of non-isomorphic factors with the same standard invariant.

Recent research in this direction has centered on Jones' planar algebra formulism. I have seen many talks by Vaughan where he says that things are hopeless at index 6 and cites the result above.

However, if one doesn't care above classifying subfactors and only wants to classify standard invariants (planar algebras) the above problem does not come up. So my questions is

How hard it is to classify (subfactor) planar algebras at index 6? And why?