Is it correct to say "do either you or John like this?", or it should be "does either you or John like this"?

Because it is possible that both "you" and "John" may like "this", I would choose "Do either ..." although I understand that "either" is one or the another (but not both). I also realise some are adamant it is "does either ..." although agreeing it sounds a little weird.

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