Polterovich's lower-bound conjecture for the Poisson bracket invariant

Let $(M,\omega)$ be a closed symplectic manifold. For a finite open cover $\mathcal{U}=\{U_i\}$ of $M$, let $e(U_i)$ denote the displacement energy of $U_i$ and set

$$e(\mathcal{U}):=\max_i e(U_i).$$

The Poisson bracket invariant $pb(\mathcal{U})$ is a non-negative invariant of the cover. Polterovich's conjecture. There exists a constant $c_M$, depending only on the symplectic manifold $(M,\omega)$, such that for every finite open cover $\mathcal{U}=\{U_i\}$ of $M$,

$$pb(\mathcal{U})\geq \frac{c_M}{e(\mathcal{U})}.$$

The conjecture gives a uniform lower bound for the Poisson bracket invariant in terms of the maximal displacement energy of the covering sets. In higher dimensions the conjecture is still open, and all known lower bounds for $pb$ decay with the degree of the cover.