Only true if you use a 'fly over' LNAV mode. If you let it lead the turn, it should have no problem maintaining the outbound track.
I used to let the automatics sort out the outbound track using LNAV, then switch to heading select once the LNAV had sussed the drift.
Then back to radio for the localiser capture.
But that was in a very ancient aeroplane upgraded to Rockwell Collins FMS800.....partially.