 The logic \(\mathsf{HS}\) has over a thousand expressively non-equivalent fragments involving only some of the modal operators, which have been studied extensively (see Della Monica et al. 2011 for a recent survey). \(\mathsf{HS}\) and most of its fragments are very expressive, and the respective notions of validity are usually undecidable (under some additional assumptions even highly undecidable, viz. \(\Pi^{1}_{1}\)-complete). However, some quite non-trivial decidable fragments of HS have been identified. Probably the best studied one is the neighborhood interval logic, which involves the operators \(\langle A \rangle\) and \(\langle\overline{A}\rangle\) (Goranko et al. 2003). One specific axiom for \(\langle A\rangle\) (and symmetrically for \(\langle\overline{A}\rangle\)) is \(\langle A \rangle\langle A \rangle\langle A \rangle p \to \langle A \rangle\langle A \rangle p,\) saying that any two consecutive right-neighboring intervals can be joined into one right-neighboring interval.