The eta-part of Kato's main conjecture for residual representations

Let $\overline{\rho}$ be an absolutely irreducible residual rank-two representation, let $\Delta=\operatorname{Gal}(\mathbb{Q}(\zeta_p)/\mathbb{Q})$, and let $\eta:\Delta\to\mathbb{Z}_p^\times$ be a character. Write $M^\eta$ for the $\eta$-component of a $\mathbb{Z}_p[\Delta]$-module, and assume the equivalent hypotheses preceding the conjecture so that the relevant quotient and $H^2$ are finite-dimensional $\mathbb{F}$-vector spaces. The eta-part of Kato's main conjecture for $\overline{\rho}$.

$$\dim_{\mathbb{F}}\left(H^1_{\mathrm{Iw}}(\mathbb{Z}[1/\Sigma_{\overline{\rho}}],\overline{\rho}^*(1))^\eta/\operatorname{Im}(\boldsymbol{z}_{\Sigma_{\overline{\rho}},1}(\overline{\rho})^\eta)\right)=\dim_{\mathbb{F}}\left(H^2_{\mathrm{Iw}}(\mathbb{Z}[1/\Sigma_{\overline{\rho}}],\overline{\rho}^*(1))^\eta\right).$$

This is the residual finite-level analogue of Kato's main conjecture. The preceding theorem establishes the finiteness needed to formulate it, but the paper does not establish the equality.