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Pulsar J1320-5359 was recently discovered to have one of the highest ever measured long-term braking indices (n and amp amp amp gt 100). This is far in excess over the canonical value of 3 that one would expect if pulsars slow down their rotation as rotating magnetic dipoles. Although some variation of n (on the order of a few) around 3 has been seen before such extremely large braking indices are a novel finding and their physical origin is unclear. A promising hypothesis suggests magnetic field decay as explanation. This can be observationally checked since magnetic field decay should also manifest itself in an unusually high thermal X-ray luminosity. Based on several properties PSR J1320-5359 is the best target to test whether the magnetic field decay hypothesis can explain the high braking indices. [truncated! Please see actual data for full text]