Polterovich's Poisson bracket conjecture for displaceable covers

Let $(M,c9)$ be a symplectic manifold. For a positive collection $\mathcal{F}$, let $pb(\mathcal{F})$ denote its Poisson bracket invariant, and for an open cover $\mathcal{U}$ let $pb(\mathcal{U})$ be the infimum of $pb(\mathcal{F})$ over positive collections subordinate to $\mathcal{U}$. Let $e_H(\mathcal{U})$ denote the displacement energy of the cover. Poisson bracket conjecture. There exists a constant $C>0$ depending only on $(M,\omega)$ such that, for every open cover $\mathcal{U}$ constituted of displaceable sets,

$$pb(\mathcal{U})e_H(\mathcal{U})\geq C.$$

The conjecture asks for a uniform positive lower bound, independent of the displaceable cover, for the product of the Poisson bracket invariant and displacement energy. The source discusses it as an open conjecture and notes that a positive constant depending on the individual cover is already known.