I think it can be said with high certainty that those are not independent, even more with the already known role of innate lymphoid cells.
One additional comment: be aware of the term "split", as the term "split tolerance" in transplant immunology world have a different meaning, i.e: regulation of the indirect alloresponse (response responsible for chronic rejection) with "paradoxical" activation of direct allorecognition (which mediates the acute rejection).