On the conservation results for local reflection principles

For a class $\varGamma $ of formulas, $\varGamma $ local reflection principle $\textrm{Rfn}_{\varGamma }(T)$ for a theory $T$ of arithmetic is a scheme formalizing the $\varGamma $-soundness of $T$. Beklemishev (1997, Theoria, 63, 139–146) proved that for every $\varGamma \in \{\varSigma _{n}, \varPi _{n+1} \mid n \geq 1\}$, the full local reflection principle $\textrm{Rfn}(T)$ is $\varGamma $-conservative over $T + \textrm{Rfn}_{\varGamma }(T)$. We firstly generalize the conservation theorem to nonstandard provability predicates: we prove that the second condition $\textbf{D2}$ of the derivability conditions is a sufficient condition for the conservation theorem to hold. We secondly investigate the conservation theorem in terms of Rosser provability predicates. We construct Rosser predicates for which the conservation theorem holds and Rosser predicates for which the theorem does not hold.