In 1990s, P. Li and L.-F. Tam studied the asymptotic Dirichlet problem on proper harmonic maps between the hyperbolic spaces, and showed an existence and uniqueness result under the $C^1$ boundary regularity. We generalize it to asymptotically hyperbolic manifolds. Analogously to Eells–Sampson’s theorem for closed manifolds and Hamilton’s theorem for compact manifolds-with-boundary, the unique existence in each relative homotopy class is shown under some assumption. This talk is based on a joint work with K. Akutagawa