I mean, their first three arguments in the filing are Contracts Clause, 5th Amendment, and 14th Amendment. They led with those.
But all that aside - and they're going to win on at least 2 of those 3 - so what if it's just a 1A issue? We're not talking about a local parking regulation. It's the First Amendment, one of the fundamental things that make us, us. And it specifcially prohibits the government from penalizing speech it doesn't like - that's the sole purpose of an entire clause in the amendment. It's one of the two reasons why we speak American, not English (the other is the French navy, obviously.)