Can we prove the consistency of $\mathsf{ZF+SVC}$ + "There is a Reinhardt cardinal?" (Preferably from the consistency of $\mathsf{ZF}$ with a Reinhardt cardinal, but using a stronger assumption is also okay.)

Here $\mathsf{SVC}$ means the Small Violation of Choice, claiming the axiom of choice is forcible by a set forcing.

Here are some easy observations.

• Let $\kappa$ be a critical point of $j\colon V\to V$. Then no forcing $\mathbb{P}\in V_\kappa$ can force $\mathsf{AC}$. Similarly, no forcing $\mathbb{P}\in V_{\kappa^\omega(j)}$ can force $\mathsf{AC}$ (as $j^n(\kappa)$ is also a critical point of some elementary embedding $V\to V$.)

• If there is a super Reinhardt cardinal $\kappa$, then $\mathsf{SVC}$ fails: if there is a $\mathbb{P}$ forcing $\mathsf{AC}$, then we can pull it back to $V_\kappa$ by using $j\colon V\to V$ satisfying $j(\kappa)>\operatorname{rank} \mathbb{P}$.

Can we prove the consistency of $\mathsf{ZF+SVC}$ + "There is a Reinhardt cardinal?" (Preferably from the consistency of $\mathsf{ZF}$ with a Reinhardt cardinal, but using a stronger assumption is also okay.)

Here $\mathsf{SVC}$ means the Small Violation of Choice, claiming the axiom of choice is forcible by a set forcing.

Here are some easy observations.

• Let $\kappa$ be a critical point of $j\colon V\to V$. Then no forcing $\mathbb{P}\in V_\kappa$ can force $\mathsf{AC}$. Similarly, no forcing $\mathbb{P}\in V_{j^\omega(\kappa)}$ can force $\mathsf{AC}$ (as $j^n(\kappa)$ is also a critical point of some elementary embedding $V\to V$.)

• If there is a super Reinhardt cardinal $\kappa$, then $\mathsf{SVC}$ fails: if there is a $\mathbb{P}$ forcing $\mathsf{AC}$, then we can pull it back to $V_\kappa$ by using $j\colon V\to V$ satisfying $j(\kappa)>\operatorname{rank} \mathbb{P}$.