Unramifiedness and nilpotent ramification conjecture for Hilbert modular cohomology

Let $F$ be a totally real number field of degree $g$ over $\mathbb{Q}$ and let $p$ be an arbitrary prime number. Fix a prime $\mathfrak{p}$ of $F$ lying above $p$, and let $\kappa_\mathfrak{p}=((k_\tau)_{\tau\in\Sigma},w)$ be a paritious weight such that $k_\tau=1$ for all $\tau\in\Sigma_\mathfrak{p}$. Let $\mathbb{T}$ denote the image of the universal tame Hecke algebra acting on

$$H^\bullet(\operatorname{Sh}^{\operatorname{tor}},E/\mathcal{O}\otimes_\mathcal{O}\omega^{\kappa_\mathfrak{p}}(-\mathtt{D})),$$

and let $\mathfrak{m}$ denote a non-Eisenstein maximal ideal of $\mathbb{T}$, with associated Galois representation $\bar\rho$. Let $R_\mathfrak{p}$ denote the universal ring for framed $\mathcal{O}$-deformations of $\bar\rho|_{G_\mathfrak{p}}$, and let $\mathscr{I}$ be the proper ideal of $R_\mathfrak{p}$ cutting out the locus of unramified lifts.

Unramifiedness and nilpotent ramification conjecture. The representation $\bar\rho$ is unramified at $\mathfrak{p}$, and there exists a positive integer $n$ depending on $g$ such that $\mathscr{I}^n$ annihilates the $R_\mathfrak{p}$-module

$$H^\bullet(\operatorname{Sh}^{\operatorname{tor}},E/\mathcal{O}\otimes_\mathcal{O}\omega^{\kappa_\mathfrak{p}}(-\mathtt{D}))_\mathfrak{m}.$$

This is motivated by the expected properties of modular Galois representations arising from forms of weight $(1,\ldots,1)$. It predicts both unramifiedness of the residual representation at $\mathfrak{p}$ and a uniform nilpotence statement for the ramified deformation directions acting on the localized cohomology. The source does not indicate whether the conjecture has been resolved.