Yes, it is the connector!
Generally speaking each of the buttons affect different electrical circuit resistance. I mean by that circuit between two pins in the middle of the connector. Two aside are power and ground for buttons backlight. You can try close the mentioned circuit by resistor. About the resistance value try 1,6 kOhm for volume down and 3 kOhm for volume up.
But there is one thing with bother me, I am not sure is the signal should be continuous or kind of short impulse. Any way, all you need is the steering with multifunction control and airbag contact with two connectors to have everything working.