Mazur–Tate's multiplicative refined BSD conjecture

Let $E/\mathbb{Q}$ have split multiplicative reduction at a prime $p$, put $G_p=\left(\mathbb{Z}/p\mathbb{Z}\right)^*/\langle-1\rangle$, and let $\lambda(a,p)$ be the even modular symbol. Let $q_p$ be the Tate period, $\tilde q_p=q_p/p^{\operatorname{ord}_p(q_p)}$, and let $\pi_\ell:G_p\to\operatorname{Syl}_\ell(G_p)$ be the Sylow projection. Let $R$ be the smallest subring of $\mathbb{Q}$ containing $\{\lambda(a,p)\}_{a\in G_p}$ and $\tau^{-1}$, where $\tau=\#E(\mathbb{Q})$ is finite, and define $S$ by $\ell\in S$ iff $\ell\mid\#G_p$ and $\ell^{-1}\notin R$. Mazur–Tate's multiplicative refined BSD conjecture. Then $\theta_p\in I(R,G_p)$ and, for every $\ell\in S$,

$$\prod_{a\in G_p}\pi_\ell(a)^{\lambda(a,p)}\equiv\pi_\ell(\tilde q_p)^{\frac{\#\Sha_E\prod_{p'\in\mathcal{P}-\{p\}}C_{p'}}{\tau^2}}.$$

This is the multiplicative reformulation of the refined Mazur–Tate conjecture for one split multiplicative prime; it is studied numerically in the source, with no resolution stated.