I will discuss the Poisson bracket invariant of a cover, which was initially introduced by L. Polterovich. The works of L. Polterovich, S. Seyfaddini and S. Ishikawa have studied this invariant via Floer theory, and lower bounds for it were established in some situations. I will explain how one can obtain the conjectural lower bound in dimension 2, using only elementary arguments. This is a joint work with A. Logunov and S. Tanny. We express our gratitude to F. Nazarov for his contribution to this work.