The gap between near commutativity and almost commutativity in symplectic category

On any symplectic manifold of dimension greater than 2, we construct a pair of smooth functions, such that on the one hand, the uniform norm of their Poisson bracket equals to 1, but on the other hand, this pair cannot be reasonably approximated (in the uniform norm) by a pair of Poisson commuting smooth functions. This comes in contrast with the dimension 2 case, where by a partial case of a result of Zapolsky, an opposite statement holds.