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 with you, some said that the popularity is about price wether it's low or high but the popularity does not depend only on price it also depends on it's usage and it's profile. If everyone know bitcoin and it's capabilities on keeping your money safe and sound and other advantages then bitcoin will be popular even though bitcoin's price is stable.