Has there been any investigation into the ATR quality of the central alveolar approximant [ɹ]? It is very vowel-like and I have this theory that it could simply be the result of an advanced tongue root and a retracted tongue tip. Is that a crazy idea? I know there is probably more to it than that.
If you are talking about [ATR] the phonological feature, then you will want to know whether there is [-ATR] approximant that alternates with [r] in some language. If you are talking about the physical parameter normally correlated with [ATR] in languages which have that feature, then it gets tricky since the exact articulatory/acoustic correlates to be associated with [ATR] are not fully agreed upon.
For the latter question, see, inter alia, Edmondson and Esling (2009) "The valves of the throat..."; Fulop, Kari and Ladefoged (1998) "An acoustic study of the tongue root contrast..."; and references therein.