  In the past decade, we have seen an unprecedented progress in our ability of
testing general relativity in the strong field regime with black hole
observations. Most studies have focused on the so-called tests of the Kerr
hypothesis: they have tried to verify whether the spacetime geometry around
black holes is described by the Kerr solution as expected in general
relativity. One can follow either a theory-specific analysis or an agnostic
approach. Each strategy has its advantages and disadvantages. In this work, we
study the ability of agnostic X-ray tests of the Kerr hypothesis to discover
new physics. We simulate X-ray observations of bright Galactic black holes of
specific theories of gravity and we analyze the simulated data with a
reflection model employing the correct theory of gravity and another reflection
model for agnostic tests of the Kerr hypothesis. Our results suggest that
agnostic X-ray tests are valid tools to discover new physics, but their
constraining power may be lower than a theory-specific analysis.
