No, I'm not in private (incognito?), only use that for other stuff... I've tried to test this, and it did remember my progress after just refreshing the site, but i think it was yesterday i tried that and today its forgotten the progress again. So it seems to happen over time. I'll try to see what happens in Chrome.
EDIT: I'm in chrome now and it seems i already quickly tested this last time, cause i started from the first checkpoint, so in that case it could look like safari is wiping the local storage after a while. Now I cleared first stage to test it so we'll see if chrome remembers that. Temporary conclusion: don't use safari for this game.
EDIT 2: Confirmed, Safari erases the data after a (very) short while, Chrome remembered most of the checkpoints in the stage, though i cleared the whole one. Firefox probably works the same way. Opera doesn't erase the data, but transforms it into a Trojan horse. Netscape asks for the floppy disk to save on. Internet Explorer crashed when opened.