Whoa whoa whoa, that's (1) not correct[1], but (2) shameless goalpost motion in any case.

The whole premise of your original contention was that we should measure like-profession salaries to see whether or not there is an effect. Then when no effect was shown, you switched it up in favor of an argument that (again, incorrectly) predicts that such an effect can't be shown at all. That's not good faith discussion.

[1] Immigrant labor is arriving, by definition, in a pre-existing market. If immigrants can't be hired more cheaply than existing labor, by definition they can't be pulling wages down.