Answer:
This is a sensitive and complicated issue which must be asked to a Rav proficient in the Halachos of geirus and personally familiar with the person.
The general rule is that a conversion is invalid even retroactively if the convert knowingly refused to accept at the time of their conversion a single Halacha or even minhag. However if they accepted everything they knew to be obligatory but were inaccurately misinformed that covering the hair is merely a stringency, the conversion would be valid and they would have the same status as any Jew.
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