'What really influences the future of people's career is their achievements.'

We've concluded in a prior thread that the sentence above is grammatically correct. Someone from another forum, however, says that career should be changed to careers. According to him, the singular 'career' would suggest that everybody shares the same career. I disagree and I think that careers would indicate that people have more than one career. What do you think?
I'd prefer "a person's career" except that you then have to decide:

whether to be sexist and use "his" for "his career"

or be pc and use "their" - which is increasing in popularity, but still scorned by language prescriptivists.

"One's achievements are the greatest influence on career prospects."

rephrases to avoid a host of grammar traps!
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