A short problem with minimal projections and biprojections

Question

Let $(N \subset M)$ be a finite index irreducible subfactor, $P=P(N \subset M)$ its planar algebra.
Notation: For $a,b \in P_{2,+}$ positive operators, then $\langle a,b \rangle$ is the biprojection they generate.

Property (F): $\forall a , b \in P_{2,+}$ min. proj., $\exists c \in P_{2,+}$ min. proj. such that $\langle c,a \rangle , \langle b,c \rangle \ge \langle a,b \rangle$
Problem: Is (F) true in general?

Remark: it's true if $P_{2,+}$ ia abelian or if $(N \subset M)$ is depth $2$ (both by Frobenius reciprocity).

Answer 1

No there is a group-subgroup counter-example.

Property (F'): $\forall p , q $ min. central proj., $\exists r $ min. central proj. such that $\langle r,p \rangle , \langle q,r \rangle \ge \langle p,q \rangle$

Lemma: Let $p$ be a min. central proj., then $\exists a \le p$ min. proj. such that $\langle p \rangle = \langle a \rangle$.

Corollary: (F) imples (F').
proof: Assume (F) and let $p$, $q$ min. central proj. and $a$, $b$ min. proj. such that $\langle p \rangle = \langle a \rangle$ and $\langle q \rangle = \langle b \rangle$. Then $\langle p,q \rangle = \langle \langle p \rangle,\langle q \rangle \rangle = \langle \langle a \rangle,\langle b \rangle \rangle = \langle a, b \rangle$. So there is $c$ checking (F), and we take $r = Z(c)$ for checking (F'). $\square$.

Finally, the property (F') is false in general: there is a group-subgroup counter-example in this post.