A friend of mine and I ran into the following question while reading about proper forcing, and have been unable to resolve it:
Definition. A cardinal $\kappa$ is supercompact if for all ordinals $\lambda$, there exists a transitive inner model $M_\lambda$ of $V$ and an elementary embedding $j_\lambda: V\rightarrow M_\lambda$ such that $crit(j_\lambda):=\min\lbrace\alpha\in ON: j(\alpha)\not=\alpha\rbrace=\kappa$ and $M_\lambda$ is closed under $\lambda$-sequences (that is, $^\lambda M_\lambda\subseteq M_\lambda$).
Now in the definition of supercompactness, the inner models $M_\lambda$ are allowed to vary with $\lambda$. We could define a large cardinal notion, uniform supercompactness, by removing this possibility: $\kappa$ is uniform supercompact if there is a single $M$ and $j$ such that $M$ is closed under all sequences of ordinal length and $j: V\rightarrow M$ is an elementary embedding with critical point $\kappa$. But in ZFC, this is not an interesting notion. First, assuming choice, $M$ is closed under sequences of ordinal length if and only if $M=V$ (since two models with the same ordinals and sets of ordinals are equal), so $ZFC\models$ "$\kappa$ is uniformly supercompact $\iff$ $\kappa$ is Reinhardt;" second, Kunen showed that Reinhardt cardinals are inconsistent with $ZFC$. (EDIT: as Joel points out, this statement doesn't actually make sense, as Reinhardt-ness is not a first-order property. One needs to pass to some extension of ZFC which can talk about proper classes directly, like Morse-Kelly set theory, which is the setting Kunen used for his proof.)
However, assuming $\neg$Choice, both of these points seem suspect. It is still open whether Reinhardt cardinals are consistent with $ZF$, and it is unclear to me that $ZF$ alone proves that uniformly supercompact cardinals are Reinhardt. So my questions are the following:
Question 1: Does $ZF\models$ every uniformly supercompact cardinal is Reinhardt?
Question 2: What is the consistency strength of $ZF+$ there exists a uniformly supercompact cardinal?
Obviously $ZF$ proves that every Reinhardt cardinal is uniformly supercompact; also obviously, Question 2 is really only interesting in lieu of a positive answer to Question 1. The most basic obstruction to a positive answer to Question 1 is the fact that given an elementary embedding $j: V\rightarrow M$ with $M$ closed under ordinal-length sequences and $crit(j)=\kappa$, the restriction of $j$ to $M$, $j\upharpoonright M: M\rightarrow M$ is not obviously (to me at least) an elementary embedding.