Polterovich's Poisson bracket invariant lower-bound conjecture

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

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

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 $M$,

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

This lower bound can be interpreted as an uncertainty principle for Poisson bracket invariants. The invariant is known to be strictly positive when the cover consists of displaceable sets, while the conjectured quantitative lower bound remains open in higher dimensions.