Wholeness Axiom and Ultimate L

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 ?

• I'm not sure how it affects your question, but there are several well-studied large cardinal axioms that are stronger than the wholeness axiom, such as rank-to-rank cardinals and others. See the upper attic in Cantor's attic: cantorsattic.info/Upper_attic. In consistency strength, the wholeness axiom is a weakening of $j:V_\lambda\to V_\lambda$, since $\langle V_\lambda,\in,j\rangle$ is a model of WA. – Joel David Hamkins Sep 23 '15 at 10:55
• Thanks Joel. Cantor's Attic is a great resource ! I modified my question in response to your comment. – Cosmonut Sep 23 '15 at 19:43

In his first paper Suitable extender models I'', Woodin shows that such an inner model would already relativize all the large cardinals up to having an elementary $j:V_λ\to V_λ$, which is much stronger than supercompactness itself.
In his second paper Suitable extender models II'' , Woodin proves even more and shows that many consistency-wise stronger hypotheses up to having a $j:L(V_{λ+1}) \to L(V_{λ+1})$ relativize to these models (Theorem 178).