Nearly-overconvergent universal eigenform conjecture

Let $\bar{\rho}$ be the residual representation parametrising the deformation space $\mathfrak{X}(\bar{\rho})$, and let $\mathcal{G}_{\bar{\rho}}^{[p]}$ be its universal eigenform. Let $f$ be a nearly-classical point of $\mathfrak{X}(\bar{\rho})$ corresponding to a modular form of prime-to-$p$ level.

Nearly-overconvergent universal eigenform conjecture. There is an affinoid neighbourhood

$$X_f=\operatorname{Max}A_f$$

of $f$ in $\mathfrak{X}(\bar{\rho})^{\mathrm{an}}$ over which $\mathcal{G}_{\bar{\rho}}^{[p]}$ is a family of nearly-overconvergent forms in the sense of Andreatta and Iovita.

Such a result would enable finite-slope $p$-adic Rankin–Selberg $L$-functions near crystalline classical points. It is explicitly presented as an optimistic conjecture and remains open.