Ash–Pollack–Stevens conjecture on classical points of higher-rank eigenvarieties

Let ${\mathcal E}(\mathcal N^\bullet)$ be the eigenvariety constructed from the cohomological complex $\mathcal N^\bullet$ for $\operatorname{GL}_n/\mathbb Q$, and let a cuspidal classical point correspond to an automorphic representation $\pi$. An automorphic representation $\pi$ is essentially self-dual if $\pi^\vee$ is isomorphic to a twist of $\pi$ by a Hecke character. Ash–Pollack–Stevens conjecture. If an irreducible component of ${\mathcal E}(\mathcal N^\bullet)$ contains a Zariski-dense subset of cuspidal classical points, then all those classical points come from essentially self-dual automorphic representations $\pi$. This predicts that Zariski-dense classical points on higher-rank eigenvariety components are constrained to the essentially self-dual locus; the source notes that classical points are not expected to be Zariski dense for $n>2$, and cites examples in slope zero supporting this phenomenon.