Good question ;-) I don't usually like capitalised names (!) myself, but in this case I was obviously in a different mood.
Sitting in front of this place, having a good time, I stumbled across the fact that you can read "LaCUEVA" (*) everywhere: on the sign at the entrance, on the menu, even printed on the different types of drinking glasses. And combined with the historic walls/cuevas you can see under the glass floor inside, I came to this conclusion.
Best regards and happy mapping!

(*) even without a space in between