Fine structure of even Galois deformation rings

For every even Galois representation in a family of prime conductor $\ell$, we prove that if $k \in \mathbb{Z}_{\geq 1}$ is maximal such that \[\ell \equiv 1 \bmod 3^k\] then the even mod $3$ deformation ring $R_S \otimes_{\mathbb{Z}_3} \mathbb{F}_3$ of level $S=\{\ell,3\}$ does not have a quotient