My guess is if you OC, you're going to be hassled, and if you show a CHP, you'll be asked to conceal...
However, that sign's wording does not say "Only Concealed Weapons". So they can't really requir you to conceal it, or immediately charge you for trespassing if you OC and have a CHP, (unless they ask you to leave and you refuse.)
However, all that said, it IS private property, and they CAN put any stipulations on access they want. If I wanted to hang a sign in my store that said, "No wearing of hats inside, and all trousers must be properly belted", it would be perfectly legal, and I could trespass ANYONE who attempted to enter in violation of those rules...
Technically, private property owners can deny access to ANYONE to their property for ANY reason. If a private property owner wants to bar someone wearing a Star of David necklace, or a crucifix, or an amber and jet necklace, or carrying a copy of the Koran or the Book of Mormon, or wearing an "Obama Joker" t-shirt, or carrying a copy of an Ayn Rand book, from their property, they may legally do so under the law. Just like you can do in your own residence.
But a government entity (local, state, federal) may not...
Until the SCOTUS issues a decisive ruling on whether or not "private property/public access" venues are required to follow the same legal guidelines as government with regards to limitations they may put on access, this will continue. Now that McDonald has put the 2A in the same legally protected category as the 1A, one of the next big fights (after we defeat the entire concept of governmental "reasonable restrictions") will be to hash out what (if any) discretion publicly-accessible private property owners have to restrict access...