I recently learned Sharifi's conjecture in this article by F.Takako and K.Kato.
According to my understanding, this conjecture states that (the conjugation invariant) of the projective limit $\displaystyle \lim_{\leftarrow_{n}}\mathrm{Cl}(\mathbb{Q}(\mu_{p^{n}})) [ p^{\infty}]$ is isomorphic, as a module on the projective limit of the Hecke algebra of level $p^{n}$, to (the conjugation invariant of) a quotient of the limit projective of $p $-adic cohomology of the modular curve $\displaystyle \lim_{\leftarrow_{n}} H^{1}(Y_{1}(p^{n}),\mathbb{Z}_{p })$.
Question If $Y_{1}(p^{n})$ is now the Siegel variety parametrizing the abelian varieties (with level condition), is it expected that (a quotient of) $ \displaystyle \lim_{\leftarrow_{n}} H^{1}(Y_{1}(p^{n}),\mathbb{Z}_{p})$ can be linked to Iwasawa module linked to a to a field extension (I don't really now what it should be, maybe a Selmer group...)?
EDIT1: As pointed out in the comments it should not be $H^{1}$ but $H^{\frac{g(g+1)}{2}}$ if $Y_{1}(p^{n})$ parameterize abelian varieties of dimension $g$.
EDIT2: Again, as noted in the comments, my question ultimately is "are their generalizations of Sharifi's conjectures to higher ranked groups?". I have to thank the user who helped me ask a coherent question in the comments.
