The Eisenstein quotient conjecture for the $G'$-cyclotomic maps

Let $H_{p,+}$ and $\widetilde H_{p,+}$ be the plus parts of the homology groups associated with the $G'$-quotient, let $I_{\infty}$ be the Eisenstein ideal acting on $H_{p,+}$, and let

$$\widetilde{\varpi}_{G'}:\widetilde H_{p,+}\to H^2(\mathbf Z[\zeta_N^{(p)},1/(Np)],\mathbf Z_p(2)),\qquad \varpi_{G'}:H_{p,+}\to H^2(\mathbf Z[\zeta_N^{(p)},1/p],\mathbf Z_p(2)).$$

Eisenstein quotient conjecture. The map $\widetilde{\varpi}_{G'}$ factors through the Eisenstein ideal, and $\varpi_{G'}$ induces an isomorphism

$$H_{p,+}/I_{\infty}H_{p,+}\xrightarrow{\sim}H^2(\mathbf Z[\zeta_N^{(p)},1/p],\mathbf Z_p(2)).$$

This is the $G'$-analogue of the conjecture for $X_1(N);$ the supplied text gives no resolution status.