It is probably premature to talk about any cryptocurrency price stabilizing. This market is still nascent and it won't be until a couple of years from now that we see it settle and prices reaching stability.
This stance is based on facts that crypto market has just started its journey and it needs to go a long distance before becoming one of the most successful and fundamental markets of the world. But even if it is going to reach that level, the price stability is not in the favor of the market.