Let $\mathcal{P}$ be an irreducible finite index-depth subfactor planar algebra. The $2$-boxes space $\mathcal{P}_{2,+}$ is equipped with the coproduct $(a,b) \mapsto a*b = \mathcal{F}(\mathcal{F}^{-1}(a).\mathcal{F}^{-1}(b))$ with $\mathcal{F}: \mathcal{P}_{2,\pm} \to \mathcal{P}_{2,\mp}$ the $1$-click rotation.

Let the adjoint map $a \mapsto \overline{a}=\mathcal{F}(\mathcal{F}(a))$ the $2$-clicks rotation (180°), and let $e_1$ be the trivial biprojection.

Using diagrams (and irreducibility), we see easily that if $a$ is a projection, then $e_1 \le a * \overline{a}$.

*Question*: Is there a (weak?) Frobenius reciprocity?

Else, is it (nevertheless) true for the minimal central projections?

By Frobenius reciprocity, you mean that for $a$, $b$, $c$ projections and $\alpha>0$:

$\alpha c \le a * b$ $\Rightarrow$ $\alpha b \le \overline{a} * c$ and $\alpha a \le c * \overline{b}$

By *weak* Frobenius reciprocity, you mean the same but on the range support (and so without $\alpha$).