  For a class $\Gamma$ of formulas, $\Gamma$ local reflection principle
$\mathrm{Rfn}_{\Gamma}(T)$ for a theory $T$ of arithmetic is a scheme
formalizing the $\Gamma$-soundness of $T$. Beklemishev proved that for every
$\Gamma \in \{\Sigma_n, \Pi_{n+1} \mid n \geq 1\}$, the full local reflection
principle $\mathrm{Rfn}(T)$ is $\Gamma$-conservative over $T +
\mathrm{Rfn}_{\Gamma}(T)$. We firstly generalize the conservation theorem to
nonstandard provability predicates: we prove that the second condition
$\mathbf{D2}$ 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.
