W-cyclicity conjecture for distributive subfactor planar algebras

Let $P$ be a distributive subfactor planar algebra, meaning that its lattice of biprojections is distributive. A subfactor planar algebra is w-cyclic when it has a biprojection generated by a minimal projection in the relevant sense.

W-cyclicity conjecture. Every distributive subfactor planar algebra is w-cyclic.

The statement is presented as a conjecture without a resolution in the supplied text. Nearby results establish w-cyclicity under additional numerical hypotheses, including for planar algebras with fewer than $32$ biprojections or index less than $32$, but do not settle the unrestricted claim.