  L. Buhovsky, A. Logunov and S. Tanny proved the (strong) Poisson bracket
conjecture by Leonid Polterovich in dimension $2$. In this note, instead of
open cover consisting of displaceable sets in their work, we consider open
cover constituted of topological discs and give a necessary and sufficient
condition that Poisson bracket invariants of these covers are positive.
