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The falsifiability criterion is not a theory, so it cannot be applied to itself (or to any other object that is not a theory).
When we refute the potential refutability of a certain theory (hypothesis), we will come to its irrefutability, that is, it does not meet the criterion of falsifiability (Popper's criterion), and therefore-the unscientific nature of such a theory (hypothesis).
Since the Popper criterion is put forward in the framework of theorizing about theorizing, attempts to refute it as a criterion for the scientific nature of a theory or hypothesis without applying it to a specific theory (hypothesis) should be based on competing hypotheses in the methodology of science.
So, yes, Popper's criterion has been questioned and refuted (see the Duhem-Quine thesis, for example). Was he himself the basis for his refutation? – no, – the theories and individual hypotheses that were considered scientific were refuted even before its justification as a criterion of scientific character. Is a scientific hypothesis about the fundamental falsifiability of a theory as a condition for the scientific validity of such a theory? – yes, because it was falsified.
That is, in any case, it is not the criterion of falsifiability itself that is subject to falsification, but either the falsifiability of a certain theory, or the hypothesis of a falsifiability criterion, as criteria for the scientific character of a certain theory or hypothesis.