Tough losing first argument (4)

I believe the answer is:

tiff

'argument' is the definition.
(I've seen this before)

'tough losing first' is the wordplay.
'tough' becomes 'stiff' (I've seen this before).
'losing first' means to remove the first letter.
'stiff' with its initial letter taken off is 'TIFF'.

(Other definitions for tiff that I've seen before include "Quarrel (colloq.)" , "Hard to lose opening argument" , "Row; argument" , "(Lovers') quarrel" , "pet" .)

I've seen this clue in The Telegraph.

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