We extend our previous work on tests of the Kerr black hole hypothesis using x-ray reflection spectroscopy and we infer observational constraints on the Johannsen deformation parameter ${\epsilon }_3$. We analyze a NuSTAR observation of the stellar-mass black hole in GS 1354–645, Suzaku data of the supermassive black holes in Ark 564, 1H0419–577, and PKS 0558–504, and some simultaneous XMM-Newton and NuSTAR observations of the supermassive black hole in MCG–6–30–15. All our measurements of ${\epsilon }_3$ are consistent with the Kerr metric and we find quite strong constraints from Ark 564 and MCG–6–30–15. We discuss the implications of our results and the next steps to improve our tests.

• Astrophysical studies of gravity
• X ray astronomy
• Accretion disk & black-hole plasma