Status of Conservative dismissed by eccentric (4)
I believe the answer is:
rank
'status' is the definition.
(I know that rank is a type of status)
'conservative dismissed by eccentric' is the wordplay.
'conservative' becomes 'C' (UK political abbreviation).
'dismissed by' suggests deleting specific letters.
'eccentric' becomes 'crank' (crank is a kind of eccentric).
'crank' with 'c' removed is 'RANK'.
'of' is the link.
(Other definitions for rank that I've seen before include "Coarse; tier" , "position in forces" , "Tier; social standing" , "It could be major row" , "The military row is coarsely overgrown" .)