The most common and most likely answer to this crossword clue is the 11 letter word PROSELYTISE
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Buddy explains!
The straight part
We know that "convert" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case convert = proselytise.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does PROSELYTISE mean?
PROSELYTISE verb- To advertise one’s religious beliefs; to convert (someone) to one’s own faith or religious movement or encourage them to do so.
- (by extension) To advertise a non-religious belief, way of living, cause, point of view, (scientific) hypothesis, social or other position, political party, or other organization; to convince someone to join such a cause or organization or support such a position; to recruit someone.