As stated in Woodin, Davis, and Rodriguez - The HOD dichotomy, a longstanding open problem in set theory is to construct a canonical inner model for supercompactness. In general there are various ways of spelling out what a “canonical inner model of large cardinal axiom A” means. The linked paper, as well as other work of Woodin, suggests the following weak thesis:
Thesis 1: If there are canonical inner models of supercompactness, at least one of them should be a weak extender model (as defined in 1).
As it is said in the last paragraph on page 16 in the linked paper, supported by Example 27, the exhibition of a weak extender model of supercompactness is not sufficient to solve the inner model problem for supercompacts. Of course, $V$ is trivially such a model. This is why Thesis 1 is a weak thesis.
Now, due to difficulties involved in spelling out in a general context what a canonical inner model is, the consensus view seems to be that a “canonical inner model of large cardinal axiom A” should be a proper-class iterate of a mouse. A mouse is a small iterable model of a sufficient fragment of ZFC + A, which is canonical because of the Comparison Lemma. In particular, it makes sense to speak of “the smallest mouse for ZFC + A”. For example $M_1^\sharp$ is the smallest mouse above a Woodin cardinal, and $M_1$ is the canonical inner model of one Woodin cardinal obtained by iterating the top measure of $M_1^\sharp$ out of the ordinals. So based these general considerations, we have:
Thesis 2: All canonical inner models of sufficiently large cardinals are iterates of mice.
The low-level versions of these theses are in harmony. For example, assuming the existence of a measurable and a sharp for measurability, a weak extender model for measurability is also an iterate of a mouse. But they seem to come into conflict around supercompactness. It is shown in 1 that if $\delta$ is extendible and $M$ is a weak extender model for the supercompactness of $\delta$, then there is no $j : M \to M$ with critical point $\geq \delta$. On the other hand, iterates of mice (through all ordinals) are, by their construction, self-embeddable with arbitrarily high critical point.
Question: How do we resolve the “paradox” and clarify the inner model problem?