The finite-level Fitting-ideal conjecture for Kato's zeta elements

Let $F/\mathbb Q$ be the fixed finite abelian extension, let $G=\operatorname{Gal}(F/\mathbb Q)$, let $I_F$ be its augmentation ideal, let $T$ be the relevant $p$-adic representation, and define $t_{c,d}=cd(c-\sigma_c)(d-\sigma_d)\in\mathbb Z_p[G]$. Fitting-ideal conjecture. One has

$$\left\{\Phi({}_{c,d}z_F)\mid \Phi\in\operatorname{Hom}_{\mathbb Z_p[G]}(H^1(\mathcal O_{F,S},T),\mathbb Z_p[G])\right\}=t_{c,d}\cdot\operatorname{Fitt}_{\mathbb Z_p[G]}^0(H^2(\mathcal O_{F,S},T)).$$

This predicts that the evaluations of the modified Kato zeta element generate the initial Fitting ideal of $H^2$ up to the explicit factor $t_{c,d}$; it is presented as a further arithmetic property and remains open in the stated generality.