Let $(N \subset M)$ be an irreducible finite index subfactor.

If its lattice of intermediate subfactors is equivalent to $B_3$ (the lattice of divisors of $n=p_1p_2p_3$ square free):

**Question**: Is there $p \in P_{2,+}(N \subset M)$ minimal central projection, generating the identity biprojection?

*Notation*: The biprojection generated by a projection $p$ is the smallest biprojection $b \ge p$.

The identity biprojection is the Jones projection $e_M$.

*Remark:* $B_3$ is distributive and the answer is yes for all the distributive lattices **if** the biprojection generated by a minimal central projection is always central; so it's true if $P_{2,+}(N \subset M)$ is abelian, for example the group-subgroup subfactors, and also for the depth $2$ case. It's also true if the lattice admits $<8$ vertices. $B_3$ is the smallest distributive lattice for which I don't know how to prove that in general.

In particular I don't know if it's true for the dual-group-subgroup subfactors $(R^G \subset R^H)$, but there is no counter-example for $B_3$ for $\vert G \vert \le 2000$ (except $512$, $768$, $1024$, $1280$).