Well, I guess I’d say that the argument against private language means that, however many false awakenings you have, there must be a base reality somewhere, because you are using language in your dreams, and that dream discourse is parasitic on the public language that you could only have learned in base reality.
My arguments in this essay can’t convince you that you aren’t dreaming at any given moment, only that there must be a state of wakefulness in which you exist in a world populated by other minds.