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). 



A short problem with minimal projections and biprojections
**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](https://mathoverflow.net/a/201693/34538). 


