They can succeed only if the "conservative" state is violating some federal law.
Keep in mind, it was the intent of the Founding Fathers that each state have their own state constitution and their own laws. It was inevitable laws in one state would be at variance with laws in another.
There are several instances, ranging from gun laws to Obamacare, where States have enacted laws in contradiction to the Federal law. If this case prevails, lawsuits looking to strike down those State laws will soon follow.