A cardinal $\kappa$ is called compactly gargantuan if there exists some elementary embedding $j: V \to M$ with critical point $\kappa$ from the Von Neumann universe into a transitive inner model $M$ such that $\forall \lambda: \; \lambda < \kappa \implies V_{\kappa_\lambda} \subseteq M \cap \kappa$, where $\kappa_0 = \kappa$, $\kappa_{\beta+1} = \min\{j(\delta): \delta \geq \kappa_\beta \land \delta \neq j(\delta)\}$ and $\kappa_\beta = \sup_{\delta \in \beta} \kappa_\delta$ for limit $\beta$.
Does anybody know, consistency-wise, where they would lie in the large cardinal hierarchy? So far, I have deduced that they are higher in consistency strength than cardinals $\mu$ so that $\mu$ are $\mu$-superstrong.