Only slim hope with this paradoxically? (3,6)

I believe the answer is:

fat chance

I'm afraid I can't even guess how this answer relates to the clue.

(Other definitions for fat chance that I've seen before include "Little prospect" , "No likelihood at all!" , "It will never happen!" , "Slender hope (odd though that sounds)" , "Slim hope, paradoxically!" .)

I've seen this clue in The Times and The Sun.

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