According to the Split-INFL hypothesis, the subject of a sentence moves from the specifier of the predicate to SpecTP to satisfy the EPP and lastly to SpecAgrSP to obtain NOM case. Is there any reason to think that an adverbial intervening TP and AgrSP would cause ungrammaticality according to any theory or framework?
I'm formulating an analysis of S-adverbs without positing that they adjoin to X' projections. S-adverbs such as probably, possibly, certainly, etc. may come immediately after the subject of a sentence, which means that they also precede items that are known to be generated in HeadIP/HeadTP such as auxiliaries, modals, aspectual markers, etc..
He probably did try escargot. He possibly will try escargot. He certainly must try escargot.
Without assuming the Split-INFL hypothesis, the adverb will have to be adjoined to I'. With the Split-INFL hypothesis, things can be kept consistent in terms of where adverbs adjoin to—only maximal XP projections, not X' projections.