The most common and most likely answer to this crossword clue is the 11 letter word CONJECTURED
I've seen this crossword clue in The Daily Telegraph
C
O
N
J
E
C
T
U
R
E
D
Buddy explains!

The straight part
We know that "Supposed" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case supposed = conjectured.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 CONJECTURED mean?
CONJECTURED verb- To guess; to venture an unproven idea.
- To infer on slight evidence; to guess at.