I personally think it would be even more popular, since people would be able to convert fiat to Bitcoin and hold them without having to worry about it losing value.
It would be used more as a real payment method than it is now.
I agree; however, we should also consider that it's not just with the stability aspect but the price point it commands. If it is stable and doesn't change value in a volatile manner but is only commanding a low price value, then it wouldn't matter because people would be urged to deal with something that they would not get a substantial profit.