Polterovich's Poisson bracket invariant conjecture

Let $(M,\omega)$ be a closed symplectic manifold, and let $\mathcal U = \{U_i\}_{i\in I}$ be an open cover of $M$ by displaceable sets. For a subordinate partition of unity $\mathcal F=\{f_i\}_{i\in I}$, define

$$pb(\mathcal U):=\inf_{\mathcal F}\max_{x,y\in[-1,1]^{|I|}}\left\|\left\{\sum_{i\in I}x_i f_i,\sum_{j\in I}y_j f_j\right\}\right\|.$$

For a displaceable set $U_i$, let $e(U_i)$ denote its displacement energy, and set

$$e(\mathcal U):=\max_{i\in I}e(U_i).$$

Polterovich's conjecture. There exists a constant $C=C(M,\omega)>0$, depending only on the symplectic manifold, such that

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

This conjecture proposes an optimal lower bound for the Poisson bracket invariant in terms of the magnitude of localization of the cover. It extends lower-bound questions for the non-commutativity of partitions of unity subordinate to displaceable covers; the supplied source does not indicate whether the conjecture has been resolved.