Non-optimal levels of some reducible mod $p$ modular representations

Let $p\ge 5$ be a prime, $N$ be an integer not divisible by $p$, ${\bar{\rho }}_0$ be a reducible, odd and semi-simple representation of $G_{\mathbb{Q},Np}$ of dimension $2$ and $\{{\mathrm{\ell }}_1,\cdots ,{\mathrm{\ell }}_r\}$ be a set of primes not dividing $Np$. After assuming that a certain Selmer group has dimension at most $1$, we find sufficient conditions for the existence of a cuspidal eigenform $f$ of level $N{\prod }_{i=1}^r{\mathrm{\ell }}_i$ and appropriate weight lifting ${\bar{\rho }}_0$ such that $f$ is new at every ${\mathrm{\ell }}_i$. Moreover, suppose $p\mid {\mathrm{\ell }}_{i_0}+1$ for some $1\le i_0\le r$. Then, after assuming that a certain Selmer group vanishes, we find sufficient conditions for the existence of a cuspidal eigenform of level $N{\mathrm{\ell }}_{i_0}^2{\prod }_{j\mathrm{\ne }{i}_0}{\mathrm{\ell }}_j$ and appropriate weight which is new at every ${\mathrm{\ell }}_i$ and which lifts ${\bar{\rho }}_0$. As a consequence, we prove a conjecture of Billerey–Menares in many cases.