The minimal Kurihara-number Tamagawa conjecture

Let $f\in S_k(\Gamma_0(N))$ be a newform with $k\ge 2$ and $p\ge 3$. Assume that $\rho_f$ has large image, that $f$ is not congruent modulo $\pi$ to any anomalous ordinary CM modular form, and that the idealistic integral $p$-adic Hodge theory described in the paper holds. Let $N_0$ be the prime-to-$p$ part of $N$, and let $C^{\pm}_{\mathrm{per}}$ denote the relevant period factor, with sign matching $(-1)^{k/2-1}$. The minimal Kurihara-number Tamagawa conjecture. One should have

$$\operatorname{ord}_{\pi}(C^{\pm}_{\mathrm{per}})+\partial^{(\infty)}(\widetilde{\boldsymbol{\delta}}^{\mathrm{min},\dagger})=\sum_{\ell\mid N_0}\operatorname{length}_{\mathcal O}\left(\frac{\mathrm{H}^1_{\mathrm{ur}}(\mathbb{Q}_\ell,W_f^\dagger)}{\mathrm{H}^1_f(\mathbb{Q}_\ell,W_f^\dagger)}\right).$$

This is the “after taking $\operatorname{exp}^*$” refinement of the Tamagawa-number conjecture. Its formulation depends on the stated integral $p$-adic Hodge-theoretic assumption, and the source gives no proof in the stated generality.