  We study the Poisson bracket invariant, which measures the level of Poisson
noncommutativity of a smooth partition of unity, on closed symplectic surfaces.
Motivated by a general conjecture of Polterovich and building on preliminary
work of Buhovsky--Tanny, we prove that for any smooth partition of unity
subordinate to an open cover by discs of area at most $c$, and under some
localization condition on the cover when the surface is a sphere, then the
product of the Poisson bracket invariant with $c$ is bounded from below by a
universal constant. Similar results were obtained recently by
Buhovsky--Logunov--Tanny for open covers consisting of displaceable sets on all
closed surfaces, and their approach was extended by Shi--Lu to open covers by
nondisplaceable discs. We investigate the sharpness of all these results.
