Does every intermediate logic between intuitionistic logic and classical logic admit a decidable axiomatization by a finite set of rule extensions of the implication-negation fragment?

The implication-negation fragment of intuitionistic logic captures reasoning about conditionals and negation without disjunction or conjunction. When one adds new inference rules to this fragment, one obtains a lattice of stronger systems sitting between pure intuitionistic logic and classical logic. The open problem is whether every such intermediate logic, when restricted to the implication-negation language, can be finitely and decidably axiomatized purely through rule extensions of this fragment, or whether there exist intermediate logics whose implication-negation fragments escape any such finite rule-based description.