You may disagree, but you are wrong. We’re not talking about what his name is, but how it is spelled. (And Giménez would be incorrect in both English and Spanish based on how he pronounces it.) What would be the guy’s “correct” name if he moved to Russia? Would it be incorrect to transliterate it into Cyrillic? What about Arabic? Is the only correct way to spell the name how it’s spelled in English in the Latin alphabet? The issue is somewhat disguised in Spanish because the alphabet it uses is so similar to that used by English, although not exactly the same. And the spelling system is even more different (that is, if English can be said to have a system). But it’s basically the same issue. The name as pronounced is spelled Gimenez in English, but it would be Gímenez in Spanish.
I often tell people here in Panama my name is Jorge instead of George to avoid misunderstanding. And my name in the US is George Richard Fatherssurname, but here it’s George Richard Fathersname Mothersmaidenname. (When people see my name written as George Richard Fathersurname, they assume Richard is my surname.)
If he moved to Argentina and some other places in South America. At least here in Panama, “y” is not equivalent to the English “j” sound. I’m not sure what spelling you might use to convey to a Panamanian the way Gimenez pronounces his name.