Sharifi's Eisenstein conjecture for the cyclotomic $K_2$ map

Let $N$ be the level, let $\Pi_N$ be the map from the plus part of modular-symbol homology to the plus part of $K_2(\mathbb{Z}[\mu_N])\otimes_{\mathbb{Z}}\mathbb{Z}'$, let $\mathbb{T}^*_N$ be the adjoint Hecke algebra, and let $I_N$ be its full Eisenstein ideal generated by $T^*_{\ell}-1-\ell\langle\ell\rangle^*$ for primes $\ell\nmid N$ and by $U^*_{\ell}-1$ for primes $\ell\mid N$. Write $\mathbb{Z}'$ for the localization used in the source and $(\cdot)_+$ for the plus part.

Sharifi's conjecture. The map $\Pi_N$ factors through a map

$$\varpi_N\colon H_1(X_1(N),\mathbb{Z}')_+\otimes_{\mathbb{T}^*_N}\mathbb{T}^*_N/I_N\longrightarrow (K_2(\mathbb{Z}[\mu_N])\otimes_{\mathbb{Z}}\mathbb{Z}')_+,$$

and this induced map is an isomorphism.

Part (a) strengthens the earlier $p$-adic Eisenstein conjecture for primes $p\mid N$. The source states that the earlier conjecture was proved by Fukaya and Kato; the status of the full factorization-and-isomorphism statement is not separately established in the supplied text.