Weight k-2 cohomological p-newness conjecture

Let $d\mid M$ with $d\neq M$, and define

$$C_{k-2,l}^{\psi,\phi}=(\mathbb{Q}_l/\mathbb{Z}_l)(3-k)(\psi^{-1}\phi).$$

Let $\mathcal{P}_{NM/d}$ denote the set of prime divisors of $NM/d$, and let $c^{”}\in H^1_{\mathcal{P}_{NM/d}}(\mathbb{Q},C_{k-2,l}^{\psi,\phi})$ be the class constructed from the weight $k-2$ congruence. Set

$$\mathcal{S}=\\{p\in\mathcal{P}_M\mid \operatorname{ord}_{\lambda'}(\psi(p)-\phi(p)p^{k-2})>0\\}.$$

Weight $k-2$ p-newness conjecture. The class $c^{”}$ is $p$-new for every $p\in\mathcal{S}$. This is motivated by the expectation that the general newform Eisenstein congruence conjecture would allow the auxiliary eigenform to be chosen new, so that its associated cohomology class should be new at all relevant primes. The assertion is not proved in the source and remains open.