Boastful conversation: hang up (3)
I believe the answer is:
gas
'boastful conversation' is the definition.
(gas can mean silly or boastful talk)
'hang up' is the wordplay.
'hang' becomes 'sag' (I've seen this in another clue).
'up' says the letters should be written backwards (in down clue: letters go upwards).
'sag' backwards is 'GAS'.
(Other definitions for gas that I've seen before include "Fumes, vapour" , "Oxygen maybe" , "Something delightful" , "Possibly noble" , "Chatter (colloq.)" .)