Remark: This question is based on an open question at the end of a paper by Hamkins, Kirmayer, and Perlmutter: "Generalizations of the Kunen Inconistency".
$HOD$ as an inner model of $ZFC$ lies between $L$ and $V$. But its nature shows a wide degree of flexibility between $L$ -like and $V$-like behaviors up to different problems.
One of the problems about any inner model of $ZFC$ is consistency of existence of a non-trivial self elementary embedding. In this direction the following facts are well known:
(a) There is no non-trivial elementary embedding from $V$ to $V$.
(b) Existence of a non-trivial elementary embedding from $L$ to $L$ is an acceptable large cardinal axiom.
Here the open question is:
Question (1): Is the existence of a non-trivial elementary embedding from $HOD$ to $HOD$ an acceptable large cardinal axiom?
I usually discuss with my colleagues about it. I believe that $HOD$ in this problem has a $L$ - like behavior and existence of a non-trivial elementary embedding from $HOD$ to $HOD$ is an acceptable (not too) large cardinal axiom but one of my friends insists that the facts shows there is no such embedding. Finally we bet on it. Here I want to know about any known partial result around the open problem. So:
Question (2): Please introduce any known fact related to the possible answer of question (1) and explain that which one of the scenarios (a) or (b) for $HOD$ seems more possible by this fact.