Testing non-circular black hole spacetime with X-ray reflection

X-ray reflection spectroscopy is a powerful tool for testing the Kerr hypothesis and probing the strong gravity regime around accreting black holes. Most tests of General Relativity (GR) assume that the spacetime around a black hole is circular, meaning the metric possesses a specific symmetry structure common to the Kerr solution. However, deviations from circularity are predicted by various modified gravity theories and non-vacuum General Relativity solutions. In this work, we test a specific non-circular metric constructed based on a locality principle, where the deviation from the Kerr spacetime is driven by the local spacetime curvature. To accurately model the reflection spectrum in this background, we implement a relativistic ray-tracing code in horizon-penetrating (ingoing Kerr) coordinates, which are favored for their ability to avoid introducing curvature singularities at the horizon in non-circular spacetimes. We apply this model to the high-quality \textit{NuSTAR} spectrum of the Galactic black hole binary EXO 1846--031. Our spectral analysis reveals a source with a high inclination angle ($\iota \approx 76^{\circ}$) and a near-extremal spin parameter ($a_* \approx 0.98$). While we identify a global minimum in the parameter space suggesting a non-zero deformation ($\ell_{\mathrm{NP}} \approx 0.12$), the 99\% confidence interval fully encompasses the Kerr limit ($\ell_{\mathrm{NP}}=0$). We conclude that the current X-ray reflection data for EXO 1846--031 are consistent with the Kerr hypothesis. This work demonstrates the feasibility of using X-ray reflection spectroscopy to constrain non-circular metrics and establishes a framework for future tests.