Failure of the Blok–Esakia Theorem in the monadic setting

The Blok–Esakia Theorem establishes that the lattice of superintuitionistic logics is isomorphic to the lattice of extensions of Grzegorczyk's logic. We prove that the Blok–Esakia isomorphism σ does not extend to the fragments of the corresponding predicate logics of already one fixed variable. In other words, we prove that σ is no longer an isomorphism from the lattice of extensions of the monadic intuitionistic logic to the lattice of extensions of the monadic Grzegorczyk logic.