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I think the evidence is clear (that you have established thoroughly and set forth with such clarity): the switch inside the sustain pedal is stuck closed. I don't think anything else would explain it.
That can be confirmed with a voltmeter, set to Resistance or Ohms, across the pedal's jack, when it is unplugged and disconnected from the Roland keyboard. The reading ought to be very low resistance (about 0 to 20 Ohms or so), and it ought not to vary as the pedal is pressed or released.
Partial disassembly of the foot switch, and spray-cleaning it out with electronic contact cleaner spray or similar stuff (note: keep that spray away from other things than the foot switch! it's not good for everything by any means)... would be the right thing to try.