One fine day head 6500nm south to "downunder opera house city", in a A340-600, we had the oppostition over take us in a B777-300ER leaving a single contrail too, coming from off center, on the left side of the tail cone. I managed to sneak a few pictures in and saw the APU inlet door left side was open, thus creating I believe the excuse for a contrail off center. It turns out that they were running the APU due to a generator failure.