Zariski density of modular points on deformation components

Let $R_x^{\chi}$ be the universal deformation ring of the non-split representation $\bar{\rho}_x$ with determinant $\chi$, and let $C$ be an irreducible component of $\operatorname{Spec} R_x^{\chi}$. An irreducible one-dimensional point means an irreducible lifting of $\bar{\rho}_x$, not necessarily one of characteristic $0$.

Zariski-density conjecture. If $C$ contains an irreducible one-dimensional point, then the set of modular points contained in $C$ is Zariski dense.

This conjecture asks whether every deformation component containing an irreducible one-dimensional lifting contains sufficiently many modular points. The text derives the analogous density statement for components of the universal pseudo-deformation space, but does not establish this assertion for $R_x^{\chi}$; it is therefore open here.