On one hand due to [Kunen's inconsistency theorem][1] it is known that within $\sf ZF$, large cardinal axioms beyond [Reinhardt][2] cardinal are inconsistent with $\sf AC$.

Also some recent results of Bagaria, Koellner and Woodin (see [here][3]) suggest that very large cardinals beyond Reinhardt (e.g. [Berkeley][4] 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][5] 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][6]).  

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? 


  [1]: https://en.wikipedia.org/wiki/Kunen%27s_inconsistency_theorem
  [2]: https://en.wikipedia.org/wiki/Reinhardt_cardinal
  [3]: http://logic.harvard.edu/blog/wp-content/uploads/2014/11/Deep_Inconsistency.pdf
  [4]: http://cantorsattic.info/Berkeley
  [5]: https://en.wikipedia.org/wiki/K%C3%B6nig%27s_lemma
  [6]: https://en.wikipedia.org/wiki/K%C3%B6nig%27s_lemma#Relationship_with_the_axiom_of_choice