Hehehe, you hit the nail on the head!
Even if the Chinese use conventional warhead on these ASBMs it can easily be interpreted/mistaken by the Americans as a nuclear missile attack, the consequences are dire if they retaliate with nuclear counter-strikes, you'll have the intercontinental ballistic nuclear missiles flying all over the world.
Besides the US already warned that if the Chinese use these missiles to sink a carrier with 5000+ american soldiers on it,
whether it is conventional warhead or not, they WILL retaliate with nuclear. These missiles at best act as a deterrence and keep a power balance in the chinese waters, if the day comes that we have to use it then it'd be time for all of us to stock up food and get into the bunkers. No one will be that stupid and let's hope we'll never live to see that day.