Sharifi's conjecture on the Eisenstein quotient and cyclotomic K-theory

Let $N$ be the prime level and $p$ a prime with $p\nmid N$. Let $H_+$ be the plus part of the modular-symbol homology group, let $I$ be the Eisenstein ideal acting on $H_+$, let $\mathcal{K}=K_2(\mathbf Z[\zeta_N,1/(Np)])\otimes_{\mathbf Z}\mathbf Z_p$, and let $J$ be the augmentation ideal of $\Lambda=\mathbf Z_p[\operatorname{Gal}(\mathbf Q(\zeta_N)/\mathbf Q)]$. For the surjection $\xi:\mathbf Z_p[(\mathbf Z/N\mathbf Z)^\times]\to H_+$, write $\sigma_a(\zeta_N)=\zeta_N^a$ and let $\langle x,y\rangle$ denote the Steinberg symbol in $\mathcal K$. Sharifi's conjecture. There is a group isomorphism

$$I\cdot H_+/I^2\cdot H_+\xrightarrow{\sim}J\cdot\mathcal K/J^2\cdot\mathcal K$$

sending $\sum_{a\in(\mathbf Z/N\mathbf Z)^\times}\lambda_a\cdot\xi([a])$ to

$$\sum_{a\in(\mathbf Z/N\mathbf Z)^\times}\lambda_a\cdot\left(\langle1-\zeta_N^a,1-\zeta_N\rangle-\frac12\cdot([\sigma_a]-1)\cdot\langle1-\zeta_N^a,1-\zeta_N\rangle\right).$$

This is the paper's main form of Sharifi's conjecture, relating the Eisenstein quotient of modular homology to the augmentation quotient of cyclotomic $K$-theory; the supplied text gives no resolution status.