[AaronBrown]

This is true. If you follow another strategy and quit after the first flip different from the previous flip (that is, if you win the first you keep playing until you lose one; if you lose the first you keep playing until you win one), your expected variance per turn is 0.54.

Looking at things another way, you can imagine a graph of 10,000 hands with a certain variance per hand. If I subdivide this series and compute the average variance per hand over each interval, I can get a big variance by picking peaks and troughs of the series, and a small variance by picking middle points.

I doubt this matters much in practice, however. There are lots of factors that go into a decision to quit. Some tend to raise variance (playing until you get some result one way or the other) and some tend to lower variance (riding streaks). Moreover, all of them hit up against limits, you run out of time or money or other players.