Do you think if BTC had a stable price which could only be altered by $0.01-0.10 it would still be as popular and as used?
It will become more popular, because larger organizations will consider to use Bitcoin to transfer payments and won't need to worry about daily price fluctuations.
It can of course, at a constant interest in Bitcoin price remains at large organizations. But people have lost interest in this.
I don't think people would lose their interest in bitcoin if the stable price will happen, maybe only a certain portion of the group will lost its interest in bitcoin as they will no longer earn by doing day trading, they will surely miss the big price fluctuation of bitcoin.