State-of-the-art structure-preserving model order reduction (MOR) methods for port-Hamiltonian (pH) systems that use system-theoretic norms focus only on approximating the input-output dynamics. Nevertheless, a pH system consists of two objects: the input-output dynamics and an energy function typically called the Hamiltonian. If we thus measure only the input-output approximation quality, then the Hamiltonian approximation is not reflected. In this talk, we take a first step towards the dual-objective minimization problem of the Pareto-optimal approximation of the input-output dynamics and the Hamiltonian by noticing that in the pH ROM, we can modify the Hamiltonian without changing the input-output dynamics. Thus, we can search for the reduced Hamiltonian that best approximates the FOM Hamiltonian for a given pH ROM. Moreover, we propose a numerical algorithm to solve the resulting minimization problem and demonstrate its applicability with numerical examples. This talk describes joint work with Paul Schwerdtner (TU Berlin) and Benjamin Unger (U Stuttgart).