Coroner beginning to doubt suspect is publicly known (2,6)
I believe the answer is:
on record
'publicly known' is the definition.
(I've seen this before)
'coroner beginning to doubt suspect' is the wordplay.
'beginning to' indicates taking the first letters.
'suspect' is an anagram indicator.
The initial letter of 'doubt' is 'd'.
'coroner'+'d'='coronerd'
'coronerd' is an anagram of 'ON RECORD'.
'is' is the link.
(Other definitions for on record that I've seen before include "officially acknowledged" , "Officially recognised" , "Officially noted" , "where needles used to be put" , "Publicly known - stated in published document" .)