Torsion-corrected multiplicative Mazur–Tate 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 and $\tilde q_p=q_p/p^{\operatorname{ord}_p(q_p)}$. Let $R$ be the smallest subring of $\mathbb{Q}$ containing $\lambda(0,1)/(2\operatorname{ord}_p(q_p))$, $\{\lambda(a,p)\}_{a\in G_p}$, and $(\#E(\mathbb{Q})_{\mathrm{Tor}})^{-1}$. Define $S$ by $\ell\in S$ iff $\ell\mid\#G_p$ and $\ell^{-1}\notin R$, and let $\pi_\ell$ be the Sylow projection. Torsion-corrected multiplicative Mazur–Tate conjecture. Then

$$\sum_{a\in G_p}\lambda(a,p)=0,$$

and for every $\ell\in S$,

$$\prod_{a\in G_p}\pi_\ell(a)^{\lambda(a,p)}\equiv\pi_\ell(\tilde q_p)^{\frac{\lambda(0,1)}{2\operatorname{ord}_p(q_p)}}.$$

If $\operatorname{rk}_{\mathbb{Z}}E(\mathbb{Q})>0$, both sides are $1$. The paper proposes this torsion-corrected variation after numerical failures of the uncorrected conjecture; it reports supporting computations but no proof.