The most common and most likely answer to this crossword clue is the 7 letter word SWOLLEN
I've seen this crossword clue in The Times
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More clues leading to the result SWOLLEN
Buddy explains!

The straight part
We know that "conceited" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case conceited = swollen.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 SWOLLEN mean?
SWOLLEN verb- To become bigger, especially due to being engorged.
- To cause to become bigger.
- To grow gradually in force or loudness.
- To raise to arrogance; to puff up; to inflate.
- To be raised to arrogance.
- To be elated; to rise arrogantly.
- To be turgid, bombastic, or extravagant.
- To protuberate; to bulge out.
- Protuberant or abnormally distended (as by injury or disease).