The statement at their website is not accurate, in my opinion. Although they do have the power to prohibit firearms on a state owned property, there is nothing in the Tennessee Code Annotated that prohibits firearms in all state buildings, across the board. Here is a relevant AG opinion:
www.handgunlaw.us/documents/agopinions/TNAGOp04_020GovProperty.pdf
What it is going to come down to is whether or not the building is posted in accordance with the law or not. They can't just ban because of a misleading sentence on their website.
That's my opinion and I'm not a lawyer. However, I'm not trusting the posting on their website until I see an actual citation from the TCA.