If you have a problem with your radio reception it is possible that your radio is shutting down power for the amplifier in your antenna. I do not know why it is, just that output is controlled by the RDS processor, which will shut it after 5seconds. So simple solution is simply emulating signal from the RDS processor. 1resistor(10k SMD 0805) need to be desoldered and then turned 90 degrees so one side is still connected to the base of the transistor and other side can be connected to ground, check pictures:
EDIT: radio will not disable 12V on antenna input if around 2.5mA is drawn from it, so 390Ohm resistor should do the trick, use 1W, it can get hot!
EDIT2: soft coding of radio will change nothing.