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Actually, it's publicly owned. The city owns it. It's privately managed. It's clearly a civic center IMO, which puts it in a sort-of gray area right now.
One of the laws which was changed specifically references such buildings, which would make you think they'd be good-to-go for carry now. But another law about posting non-parks/schools may still apply.
To make it sticky, the Bridgestone is not actually posted in any way that I can find on any doors, but as LagerHead said, they wand everyone. It's their policy to not allow firearms, but have done nothing to legally prohibit them. So the real world application is that it's not legally prohibited at Bridgestone, but you'd be arguing the finer points of law with a minimum wage "security" person.