Sufficiency of level raising for 1⊕χ_l^{k−1}

Let $\rho:\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\operatorname{GL}_2(\overline{\mathbb{F}}_l)$ be the odd semisimple representation $\mathbf{1}\oplus\chi_l^{k-1}$, with $k\geq4$ even. Let $N$ be squarefree and coprime to $l$, and suppose that $\rho$ arises from a weight-$k$ newform in $\mathcal{S}_k(\Gamma_0(N))$. For a prime $p\nmid Nl$, the level-raising condition is

$$(p^k-1)(p^{k-2}-1)\equiv0\pmod l.$$

Level-raising sufficiency conjecture. If this condition holds at $p$, then $\rho$ arises from a weight-$k$ newform in $\mathcal{S}_k(\Gamma_0(Np))$. This is the precise conjecture proposed in the paper and is open there.