Let $\ell$ be an odd prime and let $\mathcal{O}$ be the ring of integers in a finite extension of $\mathbb{Q}_\ell$, with residue field $k$. Let

\begin{align*} \rho:G_{\mathbb{Q}}\to GL_2(\mathcal{O}) \end{align*}

be a continuous representation unramified outside a finite set of primes, and let $\bar{\rho}:G_{\mathbb{Q}}\to GL_2(k)$ be its residual representation. Let $\mathcal{D}$ be the fixed-determinant global deformation problem for $\bar{\rho}$ with determinant $\chi\varepsilon_\ell$, with the minimal unramified or specified semistable local deformation condition at each prime $p\ne\ell$, and with the ordinary or finite flat local deformation condition at $\ell$. Assume that:

1. $\bar{\rho}$ is absolutely irreducible and odd; 2. $\bar{\rho}$ is modular, arising from a weight $2$ cuspidal eigenform; 3. $\det\rho=\chi\varepsilon_\ell$, where $\varepsilon_\ell$ is the $\ell$-adic cyclotomic character and $\chi$ is a finite-order character matching the nebentypus on the Hecke side; 4. $\rho$ is a deformation of $\bar{\rho}$ of type $\mathcal{D}$; 5. the local deformation rings defining $\mathcal{D}$ are the Taylor-Wiles local rings: minimal unramified or semistable away from $\ell$, and ordinary or finite flat at $\ell$; 6. for the adjoint trace-zero representation $\operatorname{ad}^0(\bar{\rho})$, the Selmer group $H^1_{\mathcal{D}}(G_{\mathbb{Q}},\operatorname{ad}^0(\bar{\rho}))$ and the dual Selmer group $H^1_{\mathcal{D}^{\perp}}(G_{\mathbb{Q}},\operatorname{ad}^0(\bar{\rho})(1))$ satisfy the Taylor-Wiles numerical criterion for the deformation problem $\mathcal{D}$, so that the Taylor-Wiles $R=\mathbb{T}$ theorem applies to $\mathcal{D}$.

Then $\rho$ is modular: there exists a weight $2$ cuspidal eigenform whose attached $\ell$-adic Galois representation is isomorphic to $\rho$ after extending scalars.