Generalized Billerey-Menares conjecture for squarefree levels

Let $k\geq 4$ and let $\ell>k+1$ be prime. Let $N=p_1\cdots p_t$ be squarefree, with distinct prime factors $p_i$, and let $\varepsilon$ be an Atkin-Lehner eigensystem for $\Gamma_0(N)$, meaning a multiplicative function on the positive divisors of $N$ with values in $\{\pm1\}$ and value $1$ at $1$. Let $\mathcal{S}_k^{(\varepsilon)}(N)$ denote the subspace of cusp forms in which the Atkin-Lehner operator $W_p$ has eigenvalue $\varepsilon(p)$ for every $p\mid N$. Generalized Billerey-Menares conjecture. The following are equivalent:

1. $\ell \mid \dfrac{B_k}{2k}\prod_{i=1}^{t}(1+\varepsilon(p_i)p_i^{k/2})$ and, for every $1\leq i\leq t$, $\ell\mid (1+\varepsilon(p_i)p_i^{k/2})(1+\varepsilon(p_i)p_i^{k/2-1})$.
2. There exist a newform $f\in\mathcal{S}_k^{(\varepsilon)}(N)$ and a prime ideal $\Lambda$ over $\ell$ in the coefficient field of $f$ such that

$$\overline{\rho}_{f,\Lambda}\simeq 1\oplus\overline{\chi}_\ell^{k-1}.$$

This refines the conjecture of Billerey and Menares on non-optimal squarefree levels of $1\oplus\overline{\chi}_\ell^{k-1}$ by incorporating Atkin-Lehner eigensystems; the source provides no resolution status.