Ordinary implies ordinary Galois representations

Let $F$) be an imaginary quadratic field, let $\mathfrak{m}$ be a non-Eisenstein maximal ideal of residue characteristic $p>2$ associated to a residual Galois representation $\overline{\rho}$, and suppose that $T_v\notin\mathfrak{m}$ for every $v\mid p$. Let $\mathcal{L}$ have parallel weight $(k,k)$ with $k\geq 2$, and let $\mathbf{T}_{Q,\mathfrak{m}}$ be the localized Hecke algebra. Ordinary $\Rightarrow$ Ordinary. There exists a continuous Galois representation

$$\rho=\rho_{\mathfrak{m}}:G_F\longrightarrow \operatorname{GL}_2(\mathbf{T}_{Q,\mathfrak{m}})$$

with the following properties: for $\lambda\notin R\cup Q\cup\{v\mid p\}\cup S$, it is unramified at $\lambda$ and the characteristic polynomial of $\rho(\operatorname{Frob}_{\lambda})$ is

$$Y^2-T_{\lambda}X+N_{F/\mathbf{Q}}(\lambda)^{k-1}\in\mathbf{T}_{Q,\mathfrak{m}}[X];$$

for $v\mid p$, $\rho|D_v$ is ordinary with eigenvalue the unit root of $X^2-T_vX+N(v)^{k-1}$; for $v\in R$ and $v\in S$, its restriction to inertia is unipotent, with the stated Frobenius characteristic polynomial at $v\in S$; for $v\in Q$, under the stated invertibility, residual unramifiedness, and condition $\phi(\operatorname{Frob}_v)\not\equiv\pm1\pmod{\mathfrak{m}}$, one has $\rho|D_v\sim\phi\epsilon\oplus\phi^{-1}$; and if $k=2$ and the level is prime to $v\mid p$, then $\rho|D_v$ is finite flat. This asserts that the ordinary Hecke-theoretic data give rise to an ordinary Galois representation with the prescribed local properties. The statement is presented as the paper's main assumption on the existence of Galois representations; no resolution status is supplied in the source.