I've bought some to play with. I also bought one of the competitors products and will take it apart to see what they've done.
Current requirement is 60mA @ 2V, plus a bit of overhead so let's say 0.1A once it's half discharged (less above the half voltage point, more above). Quite a bit for a small supercap and I have to admit, it seems unlikely it can do that for 4 minutes.
I'v read reports of these devices failing after two years, but then that's the same with li-ion.