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Relativistic magnetic reconnection studies have focused on symmetric configurations so far, where the upstream plasma has identical properties on each side of the layer. The boundary layer between a relativistic jet and an accretion flow forming around a supermassive black hole may present an asymmetric configuration in terms of plasma composition, bulk velocity, temperature and magnetization. In this work, we aim to conduct the first study of relativistic magnetic reconnection where the upstream plasma is composed of electron-positron pairs on one side, and electrons and ions on the other. We also investigate the role of a relativistic symmetric shear flow applied along the reconnecting field lines. We simulate magnetic reconnection using two-dimensional particle-in-cell simulations. The initial setup is adapted from a classic Harris layer without guide field, modified to accommodate plasma-composition and shear asymmetries in the upstream medium. For a composition-asymmetric setup, we find that the reconnection dynamics is driven by the electron-ion side, which is the plasma with the lowest magnetization. The energy partition favors accelerating ions at the expense of electrons even more than in a corresponding symmetric setup. With respect to shear, a super-Alfv\'enic upstream decreases the laboratory-frame reconnection rate, but, unlike in non-relativistic studies, does not shut off reconnection completely. The asymmetries examined in this work diminish the overall efficiency of electron acceleration relative to corresponding symmetric configurations. In the context of a black hole jet-disk boundary, asymmetric reconnection alone is probably not efficient at accelerating electrons to very high energies, but it might facilitate plasma mixing and particle injection for other acceleration channels at the interface., Comment: 10 pages, 9 figures, accepted for publication in Astronomy & Astrophysics