The most common and most likely answer to this crossword clue is the 9 letter word DISGRACED
I've seen this crossword clue in Financial Times
D
I
S
G
R
A
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D
Buddy explains!

The straight part
We know that "suffering obloquy" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case suffering obloquy = disgraced.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 DISGRACED mean?
DISGRACED verb- To put someone out of favor; to bring shame or ignominy upon.
- Having been disgraced.