Sharifi's reciprocity-map inverse conjecture

Fix an even character $\theta:(\mathbb{Z}/Np\mathbb{Z})^{\times}\rightarrow\overline{\mathbb{Q}}_p^{\times}$ and let $\omega$ be the Teichmüller character. Assume $p\nmid\varphi(M)$, $r>0$, $\theta$ is primitive, the stated restriction condition on $\theta\omega^{-1}$ holds, and, when $M=1$, $\theta\ne\omega^2$. Let $\varpi_{\theta}$ and $\Upsilon_{\theta}$ be Sharifi's maps between the indicated $\theta$-components of cyclotomic cohomology and the Eisenstein quotient of modular-curve homology. Sharifi's reciprocity-map inverse conjecture. The maps $\varpi_{\theta}$ and $\Upsilon_{\theta}$ are inverse to each other. Fukaya and Kato proved this under certain assumptions.