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?



Is there a clear-cut analogue of the strong form of Serre's Conjecture for residually reducible Galois Representations?
Billerey and Menares have studied this question for the reducible representations $\bar{\rho} = 1 \oplus \chi\_p^{k-1} $ in <https://arxiv.org/abs/1309.3717>
In this case the prime-to-$p$ part of the Artin conductor is 1, and it is not always the case that $\bar{\rho} $ arises from a cuspidal eigenform of weight $k $ and level $1$.


