Existence of Galois representations attached to torsion Hecke classes over imaginary quadratic fields

Let $F$ be an imaginary quadratic field, let $Q$ be a finite set of primes of $F$, let $\mathbf{T}_{Q,\mathfrak{m}}$ be the completion at a maximal ideal $\mathfrak{m}$ of the Hecke algebra acting on the relevant homology, and let $\overline{\rho}$ be the residual representation associated to $\mathfrak{m}$. Assume that $\mathfrak{m}$ is non-Eisenstein and associated to $\overline{\rho}$.

Existence conjecture. There exists a continuous Galois representation

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

with the stated unramifiedness, characteristic-polynomial, local inertia, decomposition-group, and ordinary or finite-flat properties at primes outside $S(\overline{\rho})\cup Q\cup\{v\mid p\}$, in $S(\overline{\rho})$, in $Q$, and above $p$.

Such representations would attach Galois representations to torsion classes in cohomology and are the prerequisite for the modularity-lifting results over imaginary quadratic fields developed in the paper. Their existence is assumed here because it is not proved by the methods of the paper.