They have grounds on the 4th and 14th amendments but maybe not the 2nd since its status of having been incorporated against the states remains unclear. "Unclear" is a euphemism for "The Supreme Court hasn't ruled yet" which is bullsh!t in and of itself, but that's what we're working with.
In light of the events, it seems pretty likely that a judge will require more specific definitions, verbiage and training and probably award damages to the plaintiffs, but that's it (on a good day).
If memory serves, the 9th Circuit is in CA and it DID incorporate 2A against the states. If this case could be positioned such that this judge must uphold that incorporation it would make it far less likely the SCOTUS would overrule it when/if it ever gets around to dealing with 2A.
Side note: People near me have been jailed and their firearms confiscated (along with their "arsenal" of 300 rounds of ammo). The cops err on the side of caution and "let the DA figure it out". That certainly works great for the state, but it costs the individual a lot, deprives him or her of freedom and indefinite separation from their property, and attaches a stigma to that person whether they're cleared of wrongdoing or not. I personally have had run-ins with cops at the range who accuse me of owning an "illegal assault rifle". I challenged the last idiot cop to arrest me and I warned him that if he did, my lawyer and I would come after him personally. Then I asked him "Would you like to see what New York State law has to say?" and I pulled my printed copies of the law out of my range bag. He packed up and left. Douche.
It pisses me off that we have to keep copies of the law with us (those of us in "special" states) to keep those who enforce - but do not know - the law at bay.