Billerey–Menares refinement of Ramanujan's congruence for square-free levels

Let $N=MN'$ be square-free and let $k\geq 4$. Let $g\in M_k(M)$ be a newform of level $M$, with Hecke eigenvalues $\lambda_p$ for $p\mid N'$. Let $\varepsilon:D(N')\to\{\pm1\}$ be a system of Atkin–Lehner eigenvalues, and let $S_k^{(\varepsilon)}(N)$ denote the subspace of $S_k(N)$ on which each $W_p$, for $p\mid N'$, has eigenvalue $\varepsilon(p)$. Billerey–Menares's conjecture. Assume that for every $p\mid N'$,

$$\lambda_p\equiv-\varepsilon(p)p^{k/2-1}(p+1)\pmod{\mathcal I},$$

where $\mathcal I$ is a prime ideal of residue characteristic $\ell$ in the field generated by the Hecke eigenvalues of all newforms in $S_k^{(\varepsilon)}(N)$, with $\ell>k-2$ and $\ell\nmid6N$. If $M=1$ and $g=E_k$, also assume that $\ell$ divides the numerator of

$$\frac{B_k}{2k}\prod_{p\mid N}(p^{k/2}+\varepsilon(p)).$$

Then there exists a newform $f\in S_k^{(\varepsilon)}(N)$ of level $N$ such that

$$f\equiv g_{N'}^{(\varepsilon)}\pmod{\mathcal I}.$$

The conditions ensure that the oldform construction $g_{N'}^{(\varepsilon)}$ becomes cuspidal and new modulo $\mathcal I$. The conjecture would follow from an appropriate integral new-subspace theory and surjectivity of reduction; the source presents it as an unproved refinement.