Hájek and Montagna proved that the modal propositional logic ILM is the logic of $\Pi_1$-conservativity over sound theories containing I$\Sigma_1$ (PA with induction restricted to $\Sigma_1$ formulas). I give a simpler proof of the same fact.
"A Simple Proof of Arithmetical Completeness for $\Pi_1$-Conservativity Logic." Notre Dame J. Formal Logic 35 (3) 346 - 354, /Summer 1994. https://doi.org/10.1305/ndjfl/1040511342