The eta-part of weak Kato's main conjecture for universal deformations

Let $\rho_x$ be a rank-two $p$-adic representation in the universal deformation family, let $\Sigma$ be a finite set of primes containing the ramified primes and $p$, and let $\eta:\Delta\to\mathbb{Z}_p^\times$ be a character for $\Delta=\operatorname{Gal}(\mathbb{Q}(\zeta_p)/\mathbb{Q})$. Write $M^\eta$ for its $\eta$-component and let $\lambda$ denote the relevant Iwasawa lambda invariant. The eta-part of weak Kato's main conjecture for $\rho_x$.

$$\lambda\left(H^1_{\mathrm{Iw}}(\mathbb{Z}[1/\Sigma],\rho_x^*(1))^\eta/\operatorname{Im}(\boldsymbol{z}_{\Sigma,1}(\rho_x)^\eta)\right)=\lambda\left(H^2_{\mathrm{Iw}}(\mathbb{Z}[1/\Sigma],\rho_x^*(1))^\eta\right).$$

This weak form retains only equality of lambda invariants rather than equality of characteristic ideals; the paper presents it as a further conjectural consequence of the same main-conjecture framework.