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?**