I have to be honest and say that I am not a good 'lost and found' psychic. But I'll give it a whirl.
I do think the ring is in the apartment. What I see is it looks stuck in between the carpet and the way. It feels wedged in there, so when you vacuum it wouldn't be sucked up. I would go along the baseboards and see if any carpet has come loose and feel around. I also see what looks like an old antique wood desk type of thing it may be around. It's actually not a desk, I'm not sure what the correct name for it is. There is a mirror attached to it. So I guess it could be a dresser or make up area.
I hope you find your ring.