Talking about the stars has reputation diminishing (4-8)
I believe the answer is:
name-dropping
'talking about the stars' is the definition.
'reputation diminishing' is the wordplay.
'reputation' becomes 'name' (name is a kind of reputation).
'diminishing' becomes 'dropping'.
'name'+'dropping'='NAME-DROPPING'
'has' acts as a link.
(Other definitions for name-dropping that I've seen before include "Boasting of acquaintance" , "Casually mentioning famous people" , "Practice of referring to famous people in order to impress" , "Referring to one''s famous friends?" , "Casual mention of the famous in order to impress" .)