Conservative Conversion - Igrot Moshe Yoreh De'ah 3:107

Rav Moshe Feinstein was asked in 1977 about a couple in Berkeley where the woman was a convert who had converted via a Conservative beit din and the husband was a Kohein. The couple was now becoming more religious and the woman was willing to undergo a second conversion, and they wanted to know what their status was.

Rav Moshe ruled that the husband had to divorce his wife since her initial conversion was definitely not valid. Even if the Conservative Rabbis who converted her observed mitzvot and asked her to do the same, the fact that she refused the accept the prohibition of marrying a Kohein invalidated her conversion from the get-go.