We perform force-free simulations for a neutron star orbiting a black hole, aiming at clarifying the main magnetosphere properties of such binaries towards their innermost stable circular orbits. Several configurations are explored, varying the orbital separation, the individual spins and misalignment angle among the magnetic and orbital axes. We find significant electromagnetic luminosities, $L\sim 10^{42-46} \, [B_{\rm pole}/ 10^{12}{\rm G}]^2 \, {\rm erg/s}$ (depending on the specific setting), primarily powered by the orbital kinetic energy, being about one order of magnitude higher than those expected from unipolar induction. The systems typically develop current sheets that extend to long distances following a spiral arm structure. The intense curvature of the black hole produces extreme bending on a particular set of magnetic field lines as it moves along the orbit, leading to magnetic reconnections in the vicinity of the horizon. For the most symmetric scenario (aligned cases), these reconnection events can release large-scale plasmoids that carry the majority of the Poynting fluxes. On the other hand, for misaligned cases, a larger fraction of the luminosity is instead carried outwards by large-amplitude Alfv{\'e}n waves disturbances. We estimate possible precursor electromagnetic emissions based on our numerical solutions, finding radio signals as the most promising candidates to be detectable within distances of $\lesssim 200$\,Mpc by forthcoming facilities like the Square Kilometer Array.
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