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.
The only reason why bitcoin is so popular is because of the price. It changes its price randomly. So if the price of bitcoin is on stable pricenof 500$ for example, would it still be popular compared today ? Thats actually the most special thing about bitcoin ..