Well it will sound better without it.
The Bose wire is just an earth going to the Bose pin on the back of the HU. It just puts the HU into Bose mode (no highs no lows it must be Bose) which flattens out the EQ on the HU so the bass & treble have very little effect. You can either just add it to the adapter or live with bass & treble controls that actually work properly.