Fine structure of even Galois deformation rings

For every even Galois representation in a family of prime conductor $\mathrm{\ell }$, we prove that if $k\in {\mathbb{Z}}_{\ge 1}$ is maximal such that $$\mathrm{\ell }\equiv 13^k$$ then the even mod $3$ deformation ring $R_S{\otimes }_{{\mathbb{Z}}_3}{\mathbb{F}}_3$ of level $S=\{\mathrm{\ell },3\}$ does not have a quotient isomorphic to ${\mathbb{F}}_3[T]\mathrm{/}(T^{2+3^{k-1}})$. As a corollary, we deduce that the global even deformation rings $R_S$ are finite and flat in the category of ${\mathbb{Z}}_3$-modules.