Also some recent results of Bagaria, Koellner and Woodin (see here) suggest that very large cardinals beyond Reinhardt (e.g. Berkeley cardinals), could be inconsistent with weaker choice principles like $\sf DC$ under some "plausible assumptions" (borrowed from Koellner's words in his lecture slides).
Now consider the Konig's Infinity Lemma which implies tree property at $\aleph_0$, the statement that every $\aleph_0$ - tree has a cofinal branch. It is not hard to see that there is a similarity between such cofinal branches in $\kappa$ - trees and the $R$-chains that $\sf DC$ produces for a binary relation $R$ on a set $X$. In the other words tree property could be considered as a kind of Axiom of Dependent Choice. (For more information see here).
Now by replacing $\sf DC$ with tree property at Bagaria, Koellner and Woodin's observation, it is natural to ask:
Question. Within $\sf ZF$, is there any inconsistency between very large cardinal axioms beyond Reinhardt cardinals and tree property at one or more regular cardinals?