Let $p$ be a prime and $\mathbb{F}$ a finite field of characteristic $p$. The theorem of Khare and Wintenberger roughly states that an irreducible, odd Galois representation $\bar{\rho}:G_{\mathbb{Q}}\rightarrow GL_2(\mathbb{F})$ which is ramified at finitely many primes lifts to a modular Galois representation associated to an eigenform of optimal weight and level. The optimal level of this form is the prime to $p$ part of the Artin conductor of $\bar{\rho}$. This was the strong form of Serre's conjecture.

Hamblen and Ramakrishna showed that the hypothesis of irreducibility may be relaxed by showing that under some conditions, a reducible and indecomposable $\bar{\rho}$ lifts to a Galois representation associated to an eigenform. However, they are not able to optimize the level of the eigenform.

My question is the following: is it expected that the optimal level is the prime to $p$ part of the Artin conductor of $\bar{\rho}$? It may be the case that it is not exactly this, and if it is not, is there an explicit counterexample to this?