Let $\mathbf{T}$ be the reduced nearly ordinary Hecke algebra of level $N$ of Hida theory for $\operatorname{GL}_{2}$ over $\mathbb{Q}$ (or more generally over a totally real field $F$). Then $\mathbf{T}$ is finitely generated over a regular ring $\Lambda$ of dimension 3. Let $\mathfrak{m}$ be a maximal non-Eisentein ideal of $\mathbf{T}$.

By patching pseudo-representations attached to algebraic modular forms, Wiles (and Hida) have constructed a two-dimensional $G_{\mathbb{Q}}$-representation $(V,\rho)$ with coefficients in $\mathbf{T}_{\mathfrak{m}}\otimes_{\Lambda}\operatorname{Frac}(\Lambda)$. This representation admits a free 1-dimensional quotient $V^{+}$ and a free 1-dimensional quotient $V^{-}$ both stable under the action of $G_{\mathbb{Q}_{p}}$. Because $\mathfrak{m}$ is non-Eisenstein, there exists a choice of basis of $V$ such that $\rho$ has values in $\operatorname{GL}_{2}(\mathbf{T}_{\mathfrak{m}})$. The lattice $L\subset V$ corresponding to this choice of basis admits a free sub-module $L^{+}=L\cap V^{+}$ of rank 1 stable under $G_{\mathbb{Q}_{p}}$. However, it is unclear to me whether $L$ admits a free rank 1 quotient stable under $G_{\mathbb{Q}_{p}}$. This is true if $\rho$ modulo $\mathfrak{m}$ is of the form
$$\rho\sim\begin{pmatrix}\chi_{1}&*\\ 0&\chi_{2}\end{pmatrix}$$
with $\chi_{1}\neq\chi_{2}$ because then $L/L^{+}$ is generated by a single element according to Nakayama lemma. However, without this hypothesis, I don`t see an obvious proof of this fact, nor have I good reasons to believe it should be true. Does anyone know for sure?