If so, lateralization cannot be the evidence of the critical period.
Critical period hypothesis - Wikipedia
For all the poverty of the stimulus argument shows, the constraints in question might indeed be language-specific and innate, but with contents quite different from those proposed in current theories of UG. Or, the constraints might be innate, but not language-specific. For instance, as Tomasello 2003 argues, children's early linguistic theorizing appears to be constrained by their inborn abilities to share attention with others and to discern others' communicative intentions. On his view, a child's early linguistic hypotheses are based on the assumption that the person talking to him is attempting to convey information about the thing(s) that they are both currentlyattending to. (Another example of an innate but non-language specificconstraint on language learning derives from the structure of the mammalian auditory system; ‘categorical perception,’ and is relation to the acquisition of phonological knowledge is discussedbelow, §3.3.4.). Another alternative is that the constraints might be learned, that is, derived from past experiences. An example again comes from Tomasello (2003). He argues that entrenchment, or the frequency with which a linguistic element has been used with a certain communicative function, is an important constraint on the development of children's later syntacticknowledge. For instance, it has been shown experimentally that the more often a child hears an element used for a particular communicative purpose, the less likely she is to extend that element to new contexts. (See Tomasello 2003:179).
Critical Period Hypothesis | Language Acquisition | …
Suppose that the primary linguistic data were impoverished in all theways that nativists claim and suppose, too, that children know a bunch of things for which there is no evidence available — suppose, as Hornstein and Lightfoot (1981:9) put it, that “[p]eople attain knowledge of the structure of their language for which no evidence is available in the data to which theyare exposed as children.” What follows from this is that there must be constraints on the learning mechanism: children do not enumerate all possible grammatical hypotheses and test them against the data. Some possible hypotheses must be ruled out a priori. But, critics allege, what does not follow from this is any particular view about the nature of the requisite constraints. (Cowie 1999: ch.8.) A fortiori, what does not follow from this is the view that Universal Grammar (construed as a theory about the structural properties common to all natural languages, per Terminological Note 2 above) is inborn.