The no-hair theorem of general relativity states that isolated black holes are characterized by three parameters: mass, spin, and charge. In this Letter we consider Kerr black holes endowed with highly magnetized plasma-filled magnetospheres. Using general relativistic kinetic plasma and resistive magnetohydrodynamics simulations, we show that a dipole magnetic field on the event horizon opens into a split monopole and reconnects in a plasmoid-unstable current sheet. The no-hair theorem is satisfied, in the sense that all components of the stress-energy tensor decay exponentially in time. We measure the decay time of magnetic flux on the event horizon for plasmoid-dominated reconnection in collisionless and collisional plasma. The reconnecting magnetosphere should be a powerful source of hard x-ray emission when the magnetic field is strong.