On any bounded strictly pseudoconvex domain, there is a canonical metric called the Cheng-Yau metric, which is a complete Kähler-Einstein metric with negative scalar curvature. Under some further assumption, Biquard found that it can be perturbed to a family of Einstein metrics of similar curvature behavior near infinity, which we call asymptotically complex hyperbolic Einstein metrics. These metrics are parametrized by CR structures on the boundary close to the standard one that are allowed to be only "partially integrable.” I will explain the fact that the asymptotic expansion at infinity of these metrics is, up to some finite order, locally determined by the associated CR structure on the boundary.