Let $\bar{\rho}_p$ be a two-dimensional irreducible local Galois representation of $Gal(\bar{\mathbb{Q}}_p / \mathbb{Q}_p)$ on a $k$-vector space, where $k$ is a finite extension of $\mathbb{F}_p$. Let $\bar{\rho}$ be a modular irreducible representation of $Gal(\bar{\mathbb{Q}} / \mathbb{Q})$ unramified outside $\Sigma$ such that restriction of $\bar{\rho}$ to $Gal(\bar{\mathbb{Q}}_p / \mathbb{Q}_p)$ is isomorphic to $\bar{\rho} _p$. Let $\mathfrak{m}$ be the maximal ideal in the Hecke algebra $\mathbb{T} _{ \Sigma}$ corresponding to $\bar{\rho}$ (where by $\mathbb{T} _{ \Sigma}$ I mean a completed Hecke algebra unramified outside $\Sigma$, i.e. a projective limit of Hecke algebras of all finite levels which are unramified outside $\Sigma$).

**Question:** Let $R(\bar{\rho}_p)^{det}$ be a local deformation ring which classifies deformations $\bar{\rho _p}$ with a given fixed central character (so this ring is of the form $\mathcal{O}[[x_1,x_2,x_3]]$). Does there exist a surjection:

$$R(\bar{\rho}_p)^{det} \twoheadrightarrow \mathbb{T} _{\mathfrak{m}, \Sigma}$$

where $\mathbb{T} _{\mathfrak{m}}$ denotes localisation at $\mathfrak{m}$? Observe that the morphism itself is provided for example by a result of Carayol on Galois representations over local rings.

**Remark:** The answer is yes if instead of $R(\bar{\rho}_p)^{det}$ we would take a global deformation ring (i.e. ring which classifies deformations of $\bar{\rho}$).

If this is not a surjection, can we say anything meaningful about the morphism above?

It might not be a surjection for every $\mathfrak{m}$ and $\Sigma$ but can we at least always find one pair ($\mathfrak{m}, \Sigma$) for a fixed $\bar{\rho}_p$ such that it is a surjection?