Fashionable greeting in Conservative circles primarily (4)
I believe the answer is:
chic
'fashionable' is the definition.
(thesaurus)
'greeting in conservative circles primarily' is the wordplay.
'greeting' becomes 'hi' (hi is a kind of greeting).
'in' is an insertion indicator.
'primarily' suggests taking the first letters.
The first letters of 'conservative circles' is 'cc'.
'hi' going inside 'cc' is 'CHIC'.
(Other definitions for chic that I've seen before include "Elegant and stylishly fashionable" , "style?" , "In vogue" , "Unwanted sounds" , "Smart and stylish, in a French kind of way" .)