<p>The asymptotic Dirichlet problem for proper harmonic maps from the hyperbolic plane into conformally compact Einstein manifolds is discussed. The formal asymptotic expansion of solutions is used to give a holographic characterization of conformal geodesics in the boundary at infinity of the target manifold. While deeply inspired by Fine and Herfray’s work on characterizing those curves through renormalized area minimization, this work highlights that the harmonic map equation alone is sufficient for obtaining such a characterization, thereby providing a more streamlined derivation. In the appendix, a method of renormalization of the Dirichlet energy of proper harmonic maps is introduced, and it is verified that proper harmonic maps that are asymptotically totally geodesic to second order, which are used for our characterization, are critical with respect to the renormalized energy.</p>