If I had responded without reading other comments I would have said that the hotter, less dense "exhaust" gas from both candles — atmosphere enriched with carbon dioxide and soot and mostly depleted of oxygen — would accumulate at the top and snuff out the taller candle first. The idea that carbon dioxide is denser and therefore sinks is intriguing, but we do not observe a carpet of pure carbon dioxide on the ground in normal environments. Carbon dioxide molecules weigh, what, 12 + 16 + 16 units, so 44, and most of the atmosphere is diatomic nitrogen, which I think is a good bit lighter per molecule. (I am refraining from outside research.) The dynamic churning of fluids in motion maintains a pretty homogenous mixure, and only extreme gases like hydrogen and helium, or chlorine as in the Bhopal incident, would not be mixed in at all levels. My prediction: the top candle goes out first. I would expect it to burn longer than ten seconds, though. May I suggest a "quiz" tag? I have some ideas to contribute.
On reflection, I realize that I did not make use of a clue mike has given us: he asked the question. If the top candle goes out first, as one might expect due to convection, it would not make a very interesting demonstration worth photographing and posing as a quiz. The shape of the glass is significant. The hot gas from the tall candle will spread out at the top and begin to cool, and momentum will carry it down the sides of the glass, cooling it further. There will be a current based on the updraft of the hot, new exhaust gas in the center which will draw the older, cooler exhaust gas down the sides. The downdraft will be focused by the neck of the glass at the bottom, where it loses momentum, collects, and snuffs out the lower candle. I still don't think the density of carbon dioxide is significant; there will still be plenty of nitrogen in the gas at the bottom. The main mechanism is creating a flow of oxygen-depleted gas toward the bottom. This could be tested by performing the demonstration using a container with a different shape.