Modularity conjecture for odd Galois representations valued in dual groups

Let $k$ be a finite field of characteristic $p$, let $G^{\vee}$ be the dual group, and let $\rho:\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to G^{\vee}(k)$ be a representation. Suppose that $\rho$ is odd, nearly ordinary of type $\mathbf{x}$ at $p$, Steinberg at every place in a finite set $S$ not containing $p$, and unramified outside $S$. Put $N=\prod_{\ell\in S}\ell$, and let $M_{\mathbf{x}}$ be the coefficient module associated with $\mathbf{x}$.

Modularity conjecture. There is a Hecke eigenclass in $H^*(\Gamma_0(N),M_{\mathbf{x}})$ that matches $\rho$.

This is a higher-rank modularity prediction: suitably local Galois representations should arise from Hecke eigenclasses in the cohomology of arithmetic groups. The source does not establish the assertion in this generality.