Are these large cardinals properties equivalent? Consider the three following large cardinal axioms:


*

*there exists a nontrivial elementary embedding $j:V\to V$.

*there exists a n.e.e. $j:V\to M$ such that $M^{j^\omega(crit(j))}\subseteq M$.

*there exists a n.e.e. $j:V_{\lambda+2}\to V_{\lambda+2}$ for some $\lambda$.


Kunen's inconsistency theorem states that, in ZFC (with replacement extended to formulas containing $j$), these three statements are false. However there is no such (known) result within ZF (with replacement sufficiently extended) alone.
Clearly (1) implies (2). I remember seeing somewhere that (2) implies (1), but I don't remember where, so I might be mistaken. Does anyone knows how these three statements relates to each other?
EDIT: this question already deals with the case (1)=>(3) 
 A: No.
This is a very recent work in progress of myself with Juan Aguilera.

Definition. We say that $\kappa$ is a Kunen cardinal if there is a nontrivial elementary embedding $j\colon V_{\lambda+2}\to V_{\lambda+2}$ with $\lambda=\sup j^n(\kappa)$.

(Sometimes it is easier to talk about the critical points of the sequence, and sometimes on $\lambda$, like in the case of $I0$ and such.)

Theorem. If $\kappa$ is a Reinhardt cardinal, i.e. the critical point of an elementary embedding $j\colon V\to V$, then $\kappa$ is the limit of Kunen cardinals.

Proof. Note that being a Kunen cardinal is a first-order property in the language of set theory. And note that a Reinhardt cardinal itself is Kunen by a fairly easy verification. 
Let $j\colon V\to V$ be a nontrivial elementary embedding, and $\kappa$ its critical point. Now, it is easy to see that the set $A\subseteq\kappa$ defined as $\{\mu<\kappa\mid\mu\text{ is a Kunen cardinal}\}$ satisfies $\kappa\in j(A)$. Therefore it follows that $A$ is in the normal measure derived from $j$, and therefore it is stationary. 
In particular, a Reinhardt cardinal is the limit of Kunen cardinals. $\square$
