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Conjectured axiomatization for adding a Wijesekera-style modality

Develop a complete axiomatization for the intuitionistic modal logic obtained by extending the language with a Wijesekera-style modal connective, and prove that it is axiomatized by adding to Lmin and to WK the two additional schemata 'p->Op and (p∨q)->((Op->q)->q).

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Background

The paper contrasts the Fischer Servi and Wijesekera approaches to intuitionistic modality. It conjectures a precise combined axiomatization when a Wijesekera-style modality is added to the current language that already contains the Fischer Servi-style connectives.

References

About (5), we conjecture that the intuitionistic modal logic obtained by the addition of a connective à la Wijesekera to our language is completely axiomatized by adding to the axioms and inference rules of Lmin - expressed in the language based on and - and WK - expressed in the language based on and - the formulas ('p->Op and (pVq)->((Op-> q)-> q).

— Intuitionistic modal logics: a minimal setting (2502.19060 - Balbiani et al., 26 Feb 2025) in Section 12 (Conclusion)