Tom Weidig has pointed out these interesting
that rely, among other things, on NASA, a Japanese satellite, Cambridge University, John Hopkins University, broad iron K life emitted near the horizons from the severely crushed matter, measurements of the black hole spin rates, and on redshifted X-rays.
We may be entering an era of precision black hole measurements that could verify classical GR more accurately than before. I am skeptical that this could tell us something about new physics but we may always end up surprised.
See the preprint for more details: