Existence of a minimal world-inconsistent congruential modal logic

Ascertain whether there exists a consistent congruential modal logic, axiomatized using only one modality, one propositional variable, and modal depth two, that is not valid on any class of basic neighborhood world frames (sets-of-worlds semantics) yet remains valid or complete for appropriate classes of basic neighborhood possibility frames.

Background

The chapter exhibits congruential modal logics that are valid on possibility frames but invalid on world frames, illustrating increased expressivity. It then poses a refined challenge: find the simplest such logic under tight syntactic constraints (single modality, single variable, depth two).

Establishing such an example would sharply delineate the boundary between world-based neighborhood semantics and possibility-based semantics at minimal syntactic complexity.