I think this may be a silly question, but here goes. Let $\rho:\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\to \mathrm{GL}_2(\overline{\mathbf{F}_p})$ be a representation; say $\rho$ *is of S-type* if it is continuous, unramified almost everywhere, and the determinant of complex conjugation is $-1$. Serre's conjecture, now a theorem of Khare-Wintenberger, states that every $\rho$ of S-type arises from some modular form $f=\sum a_n e(nz)$ in the sense that $\mathrm{tr}\rho(\mathrm{Frob}_l)=a_l\;( \mathrm{mod}\;p)$ for (almost all) primes $l$.

Question: Are S-type representations $\rho:\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\to \mathrm{GL}_2(\mathbf{Z}/p^n\mathbf{Z})$ for $n\geq 2$ also expected/known to be modular?