It's unknown if the existence of a nontrivial elementary self-embedding of $HOD$ is inconsistent with $ZFC$. While like all other large cardinal axioms it cannot be proven consistent, several other axioms asserting the existence of similar elementary embeddings like a non-trivial $j:V \rightarrow HOD$, $j:HOD \rightarrow V$, and a $j:HOD \rightarrow HOD$ which is definable in $V$ are shown to be inconsistent with $ZFC$ by generalizations of the Kunen inconsistency. There are also plausible arguments for the inconsistency of a $j:HOD \rightarrow HOD$ based on some far-reaching conjectures of Woodin. So it is suspected (at least by some) to be inconsistent.
However, it doesn't seem it's been proven that it must be at least as strong as the strongest known large cardinals axioms in $ZFC$ believed to be consistent, which $I0$ and its natural strengthenings are universally agreed to be. So how much is known about the consistency strength of the existence of a non-trivial e.e. $j: HOD \rightarrow HOD$ compared to other large cardinal axioms in $ZFC$, and least which others is it stronger than?