From what I understand:
The Wholeness Axiom(s) is/are the "ultimate axioms of infinity", bordering on inconsistency with ZFC.
Ultimate L (Completion of ZFC) attempts to extend the orderly world of constructible sets to encompass all large cardinals.
My question is: What happens when the two ultimates meet? Are they consistent ? If so, are the Wholeness Axioms the strongest large cardinal axioms consistent with Ultimate L ?
Edit: In response to Joel's comment below, I will modify my question to "What are the strongest large cardinal axioms know to be consistent with (or at least not inconsistent with) Ultimate L ?"
For instance, I believe Ultimate L => V = HOD. Are the rank-into-rank axioms consistent with this ? What about the WA's ? Anything else in the WA or rank-into-rank domain that may contradict Ultimate-L ?