So we went to the Mega Cavern today, and it was posted as soon as you drive on to the property, "Carrying Concealed Weapons Is Prohibited." Above those words is the standard "gunbuster" image, below them is the applicable Kentucky statue. Those signs are a couple feet tall and posted in multiple places throughout and at all entrances.
I asked the manager on duty if open carry was permitted. She seemed perplexed by the question, but after I explained the difference between open and concealed carry, she said no weapons of any kind no matter how carried we allowed. She was quite pleasant, so I let it drop.