Deep-rooted (4-2-3-4)
'deep-rooted' is the definition.
(thesaurus)
This is all the clue.
(Other definitions for dyed-in-the-wool that I've seen before include "(Overly) traditional" , "Unchanging in a particular belief" , "Fixed and uncompromising" , "Retain" , "Extreme or unchanging in attitude" .)
I believe this clue is a double definition.
'deep' is the first definition.
(similar in meaning)
'rooted' is the second definition.
(thesaurus)
(Other definitions for ingrained that I've seen before include "Deep-dyed" , "Dine in rag (anag.)" , "Deep-rooted" .)