Buddy of mine's grandmother always left mayo in the cabinet.
I think it's probably fine. All mayo has vinegar and salt, which bacteria don't like. Some also have lemon juice (Hellman's and Best Foods).
I think when people get sick, it's probably more often from things besides the mayo itself. Whether the other proteins in a mayo based salad, or other food that ends up in the jar from double dipping (the squeeze bottle would help with that).
I refrigerate because I prefer mayo to be cold, but I think you can get away with.
Plus, I only use Duke's, which doesn't have lemon juice.