A famous Łoś-Tarski preservation theorem is that first-order (FO) sentences that are preserved under substructures (resp. superstructures) are precisely the universal (resp. existential) first-order sentences. It is known that some analogous of Łoś-Tarski theorem hold for many sublogics of FO (e.g. modal logics, guarded fragment, guarded negation fragment [1,2] ) but not for all of them, e.g. the two-variable first-order logic with equality [3].

Is there any known equality-free (meaning that the use of = relation is forbidden) fragment of FO for which the analogous of Łoś-Tarski fails?

[1] Modal Languages and Bounded Fragments of Predicate Logic

[2] Some Model Theory of Guarded Negation

[3] On Preservation Theorems for Two-Variable Logic