Sharp quantum reverse Young inequality for irreducible planar algebras

Let $\mathscr{P}_{\bullet}$ be an irreducible planar algebra with index $\delta^2$, where $\delta>0$. For positive operators $x,y\in\mathscr{P}_{2,\pm}$ and parameters $0<r,s,t\leq1$ satisfying

$$1+1/r=1/s+1/t,$$

write $\ast$ for the planar-algebra convolution and $\|\cdot\|_u$ for the corresponding norm. Sharp quantum reverse Young inequality. One has

$$\|x\ast y\|_r\geq\delta^{-1}\|x\|_s\|y\|_t.$$

This is presented as a sharp form of quantum reverse Young's inequality on subfactor planar algebras. The surrounding text gives a different previously proved bound, with factor $\delta^{1-2/r}$, and explicitly remarks that this bound is not sharp; the supplied span asserts the sharper factor $\delta^{-1}$, but no resolution status is provided.