Kato's main conjecture without the p-adic zeta function

Let $f$ be the modular form under consideration, let $G_ fty$ be the relevant Galois group, and let $\mathcal{O}$ be the coefficient ring. For $\natural\in\{\sharp,\flat\}$ and $0\leq i\leq p-2$, let $\mathcal{X}^{\natural}(f)^{\omega^i}$ denote the $\omega^i$-isotypic component of the Pontryagin dual of the corresponding sharp or flat Selmer group, and let $L_p(f,\natural,\omega^i,X)$ and $\xi_{i,\natural}$ be the associated $p$-adic $L$-function and correction factor. Kato's main conjecture without the $p$-adic \zeta function. There is an equality of $\mathcal{O}\llbracket G_\infty\rrbracket$-ideals

$$\mathrm{Char}_{\mathcal{O}\llbracket G_\infty\rrbracket}\left(\mathcal{X}^{\natural}(f)^{\omega^i}\right)=\left(\frac{L_p(f,\natural,\omega^i,X)}{\xi_{i,\natural}}\right).$$

This is the formulation of Kato's main conjecture that omits the $p$-adic zeta function; the statement concerns the characteristic ideal of each sharp or flat Selmer component and its relation to the corresponding $p$-adic $L$-function. The supplied text gives no resolution, so the conjecture remains open.