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There are no gravitons in General Relativity, and black holes don't evaporate in General Relativity (i.e. there is no Hawking radiation). The consequence of Hawking radiation, or something similar, being empirically verified, would be an empirical refutation of General Relativity, precisely because it *doesn't* predict this. Specifically, General Relativity doesn't predict that black holes will "evaporate" at all.

There's a semi-implicit subquestion of whether Hawking radiation predicts that some of the energy radiated away would appear as gravitational waves. I don't know, but it seems plausible. Either way, General Relativity doesn't predict it.

If Hawking radiation occurs, models of a black hole in General Relativity would be like models of an iron beam in solid mechanics. Iron beams corrode but the solid mechanics model isn't going to predict that nor the change in the relevant mechanical properties, because it doesn't model chemistry.

Could we adapt General Relativity into a different, but still non-quantum, theory that did predict this? Probably, in the same way we could have the iron beam's mechanical properties change over time "just because" from the model's perspective. My impression is any such theory would be presented as and, even more, received as an approximation to an underlying quantum gravity theory and not as a new foundational theory of gravity.