In my L1020 running the same build (10586.63), FM Radio works only when the headset is plugged in... but this time it plays the radio through the phone speakers. Disconnecting the headset kills the FM radio due to poor signal reception (first it plays some static, then dies half a second later).
Presumably, since the headset wires serve as the antenna, if the reception is good enough in an area such that an antenna is not needed in order to have good reception, then theoretically the FM radio should work even without the headset plugged in.