Mazur–Tate's refined class-group formula conjecture

Let $E/\mathbb{Q}$ be an elliptic curve, let $G_M=\left(\mathbb{Z}/M\mathbb{Z}\right)^*/\langle-1\rangle$, and let $\theta_M=\sum_{a\in G_M}\lambda(a,M)[a]$ be the Mazur–Tate element. Let $I(R,G_M)$ be the augmentation ideal and $Q_r(R,G_M)=I(R,G_M)^r/I(R,G_M)^{r+1}$. Fix a finite set $S_m$ of split multiplicative primes, set $M=\prod_{p\in S_m}p^{e_p}$ and $r=\#S_m$, and view each normalized Tate period $\tilde q_p$ as an element of $G_M$. Mazur–Tate's refined class-group formula conjecture. Assume that $\tau=\#E(\mathbb{Q})$ is finite. If $R$ is the smallest subring of $\mathbb{Q}$ containing $\tau^{-1}$ and $\{\lambda(a,M)\}_{a\in G_M}$, then $\theta_M\in I(R,G_M)^r$ and, for the image $\tilde\theta_M$ in $Q_r(R,G_M)$,

$$\tilde\theta_M=\left(\prod_{p\in S_m}[\tilde q_p]-[1]\right)\frac{\#\Sha_E\prod_{p\in\mathcal{P}-S_m}C_p}{\tau^2}.$$

Here $\Sha_E$ is the Tate–Shafarevich group and $C_p=[E(\mathbb{Q}_p):E^0(\mathbb{Q}_p)]$. This conjecture refines the BSD formula by relating the leading augmentation-ideal term of the Mazur–Tate element to arithmetic invariants; the source reports numerical study rather than a proof or disproof.