The most common and most likely answer to this crossword clue is the 10 letter word MINIMALIST
I've seen this crossword clue in The Guardian
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More clues leading to the result MINIMALIST
Buddy explains!

The straight part
We know that "eschewing ostentation" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case eschewing ostentation = minimalist.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 MINIMALIST mean?
MINIMALIST noun- One who believes in or seeks a minimal state; one who seeks to minimize or reduce to a minimum.
- Believing in or seeking a minimal state; seeking to minimize or reduce to a minimum.