Well, if we're betting, I'm betting not. A little skirmish here or there, but not a war.
The leaders of all three countries are not nuts. They've lived under the threat of nuclear holocaust for over a half-century and they don't want it. I see no indication that any of them is like the loons of N. Korea or Iran.
Again, if we're betting, my guess is that the country most likely to face a real nuclear threat is N. Korea -- which could happen if they get trigger-happy with our navy or our defense forces in S. Korea.