Cohomological p-newness conjecture for Eisenstein congruence classes

Let $N,M\geq 1$ be fixed coprime square-free integers, let $\psi$ and $\phi$ be characters of conductors $u,v\geq 1$ with $uv=N$, and let $\lambda'$ be a prime above $l$. Define

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

Let $\mathcal{P}_{NM}$ denote the set of prime divisors of $NM$, and let $c'\in H^1_{\mathcal{P}_{NM}}(\mathbb{Q},C_{k,l}^{\psi,\phi})$ be the class supplied by the Eisenstein congruence. Set

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

Cohomological p-newness conjecture. The class $c'$ is $p$-new for every $p\in\mathcal{S}$, meaning that it does not come from the corresponding relaxed Selmer group with the condition at $p$ removed. The conjecture predicts that the cohomology class arising from the congruence records the newness at every prime whose local Euler factor contributes to the congruence. The source presents this as a conjecture after noting that the preceding conditions guarantee only $p$-newforms separately, rather than a genuine newform; no general proof is given.