Conjectured non-finite axiomatizability of certain modal fragments

Show that some box-free fragments or some diamond-free fragments of the intuitionistic modal logics introduced in the paper are not finitely axiomatizable.

Background

The authors suggest studying fragments of their systems obtained by restricting to formulas without one of the modal operators. They conjecture that at least one of these kinds of fragments (box-free or diamond-free) cannot be captured by a finite axiom system.