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We extend our previous work on tests of the Kerr black hole hypothesis usingX-ray reflection spectroscopy and we infer observational constraints on theJohannsen deformation parameter $\epsilon_3$. We analyze a NuSTAR observationof the stellar-mass black hole in GS 1354-645, Suzaku data of the supermassiveblack holes in Ark 564, 1H0419-577, and PKS 0558-504, and some simultaneousXMM-Newton and NuSTAR observations of the supermassive black hole inMCG-6-30-15. All our measurements of $\epsilon_3$ are consistent with the Kerrmetric and we find quite strong constraints from Ark 564 and MCG-6-30-15. Wediscuss the implications of our results and the next steps to improve ourtests.

Constraining the Johannsen deformation parameter $\epsilon_3$ with black hole X-ray data