Sharifi's Eisenstein-ideal annihilation conjecture

Let $\widetilde{\varpi}:H_1(X_1(N),C_{\infty},\mathbf Z_p)_+\to\mathcal K$ be the cyclotomic map, and let $\mathfrak I_{\infty}$ be the Eisenstein ideal acting on the source. Sharifi's Eisenstein-ideal conjecture. One has

$$\widetilde{\varpi}(\eta x)=0$$

for every $\eta\in\mathfrak I_{\infty}$ and every $x\in H_1(X_1(N),C_{\infty},\mathbf Z_p)$. Equivalently, the map factors through the Eisenstein quotient. The conjecture is presented as Sharifi's conjecture in the paper; the supplied text gives no resolution status.