McCallum–Sharifi's surjectivity conjecture for the cyclotomic cohomology map

Let $p\nmid N\phi(N)$, let $\theta$ be a primitive even Dirichlet character of conductor $Np$, and let $\varpi_{\theta}$ denote the $\theta$-component of Sharifi's homomorphism from the minus part of modular symbols to cyclotomic cohomology. Write $H^-_{\theta}(1)$ and $S_{\theta}$ for the corresponding $\theta$-eigenspaces, $I^*_{\theta}$ for the $\theta$-component of the Eisenstein ideal, and $\Lambda_{\theta}$ for the corresponding Iwasawa algebra. McCallum–Sharifi's conjecture. The homomorphism $\varpi$ induces a surjective homomorphism of $\Lambda_{\theta}$-modules

$$\varpi_{\theta}:H^-_{\theta}(1)/I_{\theta}^*H^-_{\theta}(1)\twoheadrightarrow S_{\theta}.$$

This conjecture concerns the relationship between modular symbols and cyclotomic cohomology and is part of Sharifi's proposed connection between Eisenstein quotients and Iwasawa modules. The source records no resolution of this surjectivity statement.