Let $\bar\rho$ be an odd, absolutely irreducible, 2-dimensional mod $p$ representation of $\operatorname{Gal}(\overline{\mathbf{Q}} / \mathbf{Q})$ (with coefficients in some finite extension $k / \mathbb{F}_p$). Then one can form the universal deformation ring $R_\bar\rho$ of $\bar \rho$, parametrizing equivalence classes of lifts of $\bar \rho$ to characteristic 0 with some fixed Artin conductor away from $p$ (but no local hypotheses at $p$). Under some mild hypotheses this ring is known to be isomorphic to a power series ring in three variables over $W(k)$.

There are some results ("big R = big T" theorems), due to Boeckle, Emerton and others, which show that (under suitable hypotheses on $\bar\rho$) this ring $R_{\bar \rho}$ is canonically isomorphic to a localization of a Hecke algebra, acting on a suitable inverse limit of the cohomology of modular curves of $p$-power levels. See e.g. section 3.3.1 of Emerton's ICM survey.

In this setting, can one also exhibit the free rank 2 $R_{\bar\rho}$-module $M$ that realizes the universal deformation $$\rho^{\mathrm{univ}} : \operatorname{Gal}(\overline{\mathbf{Q}} / \mathbf{Q}) \to \operatorname{GL}(M)$$ as a quotient of some module built from cohomology of modular curves?