  The intuitionistic modal logics considered between Constructive K (CK) and
Intuitionistic K (IK) differ in their treatment of the possibility (diamond)
connective. It was recently rediscovered that some logics between CK and IK
also disagree on their diamond-free fragments, with only some remaining
conservative over the standard axiomatisation of intuitionistic modal logic
with necessity (box) alone. We show that relational Kripke semantics for CK can
be extended with frame conditions for all axioms in the standard axiomatisation
of IK, as well as other axioms previously studied. This allows us to answer
open questions about the (non-)conservativity of such logics over
intuitionistic modal logic without diamond. Our results are formalised using
the Coq Proof Assistant.
