Consequence in $\mathcal{L}_{\infty\lambda}$

Question

Consider the infinitary first-order language $\mathcal{L}_{\infty\lambda^+}$ whose non-logical vocabulary consists of $\lambda \geq \omega$ individual constants and countably many predicate constants of finite arity (i.e., the language in which we can form conjunctions of arbitrarily large sets of formulas and generalize on sets of variables whose cardinality does not exceed $\lambda$). Define logical consequence in the usual way. Let $\rho$ be a permutation of the class of sentences of $\mathcal{L}_{\infty\lambda^+}$ such that (i) $\rho(\varphi) = \varphi$ for all $\varphi$ containing only finitely many individual constants, and (ii) $\Gamma \vDash \Delta$ iff $\rho(\Gamma) \vDash \rho(\Delta)$ for all sets of sentences $\Gamma$ and $\Delta$. Does it follow that $\vDash\varphi\leftrightarrow \rho(\varphi)$ for all sentences $\varphi$ of $\mathcal{L}_{\infty\lambda^+}$? (I.e., is the entailment profile of a sentence pinned down by its entailment relations to sentences containing only finitely many individual constants?)