Weak Poisson bracket conjecture for displaceable covers

Let $(M,\omega)$ be a symplectic manifold of dimension $2n$, and let $\mathcal{U}$ be an open cover constituted of displaceable sets. For $pb(\mathcal{U})$ as above and the symplectic volume $\operatorname{Vol}(M,\omega)$, weak Poisson bracket conjecture. There exists a constant $C>0$ depending only on $(M,\omega)$ such that, for every such cover,

$$pb(\mathcal{U})\operatorname{Vol}(M,\omega)^{1/n}\geq C.$$

This weaker conjecture asserts that $pb(\mathcal{U})$ is uniformly bounded away from zero on covers by displaceable sets. The source presents it as a related, simpler conjecture and explains that the stronger conjecture would virtually imply it when $e_H(\mathcal{U})$ is sufficiently small.