Stability conjecture for near-biprojections in subfactor planar algebras

Suppose $\mathscr{P}_{\bullet,\pm}$ is an irreducible subfactor planar algebra. For $x\in\mathscr{P}_{2,\pm}$, let $\mathfrak{F}(x)$ denote its Fourier transform, and let $P,Q$ be projections and $\lambda$ a constant. Stability conjecture. For any $\varepsilon>0$, there exists $\varepsilon'$ such that if

$$\|x-P\|_2<\varepsilon'\quad\text{and}\quad\|\mathfrak{F}(x)-\lambda Q\|_2<\varepsilon',$$

then there is a biprojection $B$ such that $\|x-B\|<\varepsilon$. This asks whether simultaneous closeness to a projection and, after Fourier transform, to a scalar multiple of a projection forces closeness to a biprojection; the supplied text does not indicate whether the conjecture has been resolved.