When it is said that Kunen inconsistency theorem proves that given $\sf ZFC$ no elementary embedding can exist from the universe to itself. Most references quote full choice in stating that result, and contemplate a salvage by forsaking choice altogether like in Rheinhardt's cardinals setting.

Is there a known weaker form of choice, like dependent or countable choice, that can evade this theorem?

When it is said that Kunen inconsistency theorem proves that given $\sf ZFC$ no elementary embedding can exist from the universe to itself. Most references quote full choice in stating that result, and contemplate a salvage by forsaking choice altogether like in Reinhardt's cardinals setting.

Is there a known weaker form of choice, like dependent or countable choice, that can evade this theorem?