Purple and white wires appear to be color code used for each electric slide out controlled via your BCM... you’ve swapped the relay at BCM for the circuit to that motor..
I’d try putting a multi meter on the plus and negative wire connection terminals at the BCM for that specific slide motor and operate using override switch
If the over ride switch isn’t applying 12 volts to that circuit then you know problem is at the BCM...
https://in-command.net/wp-content/up...ting-guide.pdf
However.. if you have voltage then you know issue is downstream toward slide actuator motor.
The purple wires should be labeled 1,2,3,4, etc so you may be able to pick up a circuit identifier label on the lead to that actuator.