Depends on how it's set up. While it's true that a litre of LPG has something like 10% less energy available compared to a litre of petrol, a properly set up system will do some decent ignition timing advance to make it run more efficiently, compensating for this lot.
In theory, with some sort of aftermarket management (not sure if this even exists yet) and a good map, you should be able to get better performance on LPG than on petrol, but now we're getting into pretty specialist stuff that's likely the domain of race teams and the like. An old BMW with an aftermarket conversion wont be doing this.
You still wont lose performance though, because as soon as you mash the loud pedal it will switch back to petrol.