The Kato-system Tamagawa conjecture

Let $f\in S_k(\Gamma_0(N))$ be a newform with $k\ge 2$ and $p\ge 3$, and suppose that $\rho_f$ has large image. Let $N_0$ be the prime-to-$p$ part of $N$. The Kato-system Tamagawa conjecture. One should have

$$\partial^{(\infty)}(\boldsymbol{\kappa}^{\mathrm{Kato},\dagger})=\sum_{\ell\mid N_0}\operatorname{length}_{\mathcal O}\left(\frac{\mathrm{H}^1_{\mathrm{ur}}(\mathbb{Q}_\ell,W_f^\dagger)}{\mathrm{H}^1_f(\mathbb{Q}_\ell,W_f^\dagger)}\right).$$

This is the “before taking $\operatorname{exp}^*$” refined Tamagawa prediction. The authors explain that proving it is expected to require the Iwasawa main conjecture together with integral $p$-adic Hodge theory, and no proof is supplied.