Extendible and enhanced supercompact cardinals

Question

The paper "The large cardinals between supercompact and almost-huge" (2013) by Norman Perlmutter makes the following definition:

A cardinal $\kappa$ is enhanced supercompact if and only if there exists a strong cardinal $\theta \gt \kappa$ such that for every cardinal $\lambda \gt \theta$, there exists a $\lambda$-supercompactness embedding $j : V \to M$ such that $\theta$ is strong in $M$.

The follow-up paper "On extensions of supercompactness" (2015) by Robert Lubarsky and Norman Perlmutter proves that enhanced supercompact cardinals and extendible cardinals are hypercompact limits of hypercompact cardinals and that, if $\kappa$ is extendible and there is a strong cardinal greater than $\kappa$, $\kappa$ is an enhanced supercompact limit of enhanced supercompact cardinals.

Is the existence of an extendible cardinal, not necessarily with a greater strong cardinal, stronger or weaker than the existence of an enhanced supercompact cardinal?

Answer 1

Suppose that there is a $\lambda$-supercompactness embedding $j : V \to M$ (that is, $M^\lambda \subset M$) with critical point $\kappa$ such that $\theta \lt \lambda$ is strong in both $V$ and $M$. By the $\lambda$-closure of $M$, $j \upharpoonright V_\theta \in M$, and $j \upharpoonright V_\theta$ witnesses that $M\vDash \text{$\kappa$ is $\theta$-extendible}$. For any $\gamma \lt \theta$, the $\gamma$-extendibility of $\kappa$ in $M$ is absolute to $V_\theta$ (which is the true $V_\theta$ since $M^\lambda \subset M$) since $\theta$ is strong, and thus $\Sigma_2$-reflecting, in $M$. Thus, $\kappa$ is extendible in $V_\theta$ and, by the usual reflection argument using $j$, $\kappa$ is a limit of cardinals that are likewise extendible up to their respective next strong cardinal.