The distributive biprojection lattice conjecture for subfactor planar algebras

A subfactor planar algebra is an invariant of a subfactor, and its $2$-box space contains distinguished projections called biprojections; these form a lattice under the planar-algebraic order. A planar algebra is w-cyclic when some minimal $2$-box projection generates the identity biprojection.

Distributive biprojection lattice conjecture. Every irreducible subfactor planar algebra with a distributive biprojection lattice is w-cyclic.

The conjecture extends the theorem for central biprojections, and is known when the lattice has fewer than $32$ elements, or equivalently in the stated index range. Its general case remains open.