The classical-points characterization of partially de Rham families

Fix a prime $u mid p$, a nonempty proper \subset $J e u$ of the set $u$-adic embeddings, an integer $w$, and integers $k_{\tau}2$ with $k_{\tau} w\pmod{2}$ for every $\tau otin J$. Let $cal E(\underline{k}_J,w)'$ be the closed subspace of the eigenvariety consisting of points with weights $k_{\tau}$ for $\tau otin J$ and weight $w$ in the remaining direction, and let $cal E(\underline{k}_J,w)$ be the Zariski closure of its classical points. For $z otin cal E(\underline{k}_J,w)'(\overline{E})$, write $ho_z$ for the associated absolutely irreducible $\operatorname{Gal}_F$-representation and $ho_{z, u}:=ho_z|_{\operatorname{Gal}_{F_{ u}}}$. Partially de Rham family conjecture. The point $z$ belongs to $cal E(\underline{k}_J,w)(\overline{E})$ if and only if $ho_{z, u}$ is $J$-de Rham. This gives a geometric characterization of the Zariski closure of classical points in terms of local $J$-de Rhamness, relating the geometry of partially de Rham eigenvariety families to local-global compatibility. The source provides no resolution status for this assertion.