Presume.
As I have said many times, dictionaries record how words are used rather than how they ought to be used. I'm quite clear that "can" is an ability whereas "may" is a permission. My logic is that if two words are precisely synonymous, there's no need to have two words, hence my thesis that there are no synonyms. I can eat my own peanuts in the tea room, it would be difficult to prevent, but I wouldn't be welcome to do so.
The problem is that, in general, people don't think that deeply and don't appreciate the fine distinction between words that seem to be synonymous, which is the theme of AvP. The dictionary makes very little distinction between "curious" (as an adverb) and "inquisitive" - but if I want an adverb I'll use "inquisitive" and if I want an adjective I'll use "curious". Without that distinction it is impossible to determine the meaning of a phrase such as "mice are curious creatures": is a mouse a 'curious creature', or is it inquisitive? (Okay, it's probably both, but that's not the point.)
"Bought" isn't very good either. If somebody with learning had drafted the notice, it would have read: "Only food and drink purchased on the premises may be consumed here". These things are forgivable when spoken, less so when written for public display.