It is possible to hack the pcb to fix it. Extract the board, locate Q12, and solder a wire that shorts its collector (middle terminal) to ground. This will make Flip 4 think there is always audio going into aux.
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You may also want to remove 33-ohm resistors, that may reduce distortion for signal sources that have high-ish output impedance.
This hack causes a few side effects though: Bluetooth pairing sounds will not be played. Bluetooth audio does work, just no pairing sounds.