The canonical-period Tamagawa conjecture for Kurihara numbers

Let $f\in S_k(\Gamma_0(N))$ be a newform with $k\ge 2$ and $p\ge 3$, with $p\nmid N$ and $\rho_f$ of large image. Suppose that $f$ is ordinary at $p$ and $p$-distinguished, or that $2\le k\le p-1$. Let $\widetilde{\boldsymbol{\delta}}^{\mathrm{can},\dagger}$ be the collection of Kurihara numbers for $f$ at $s=k/2$, normalized by canonical periods. The canonical-period Tamagawa conjecture. One should have

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

This is a refined Tamagawa-number prediction, comparing an analytic fudge factor with local Tamagawa contributions away from $p$. The paper presents it as a conjecture because precise control of the relevant integral comparison maps is unavailable in this setting.