Mazur–Tate's order-of-vanishing and weak main conjectures

Let $E$ be an elliptic curve over $\mathds{Q}$, let $K$ be an abelian number field of conductor $m$, and set $G=G_K=\operatorname{Gal}(K/\mathds{Q})$. Let $\mathcal{R}$ be a ring such that $\mathcal{R}[G]$ contains the Mazur–Tate element $\theta_K^\mathrm{MT}$. Write $I_{\mathcal{R},G}$ for the augmentation ideal of $\mathcal{R}[G]$, $S(E/K)$ for the integral Selmer group, $r=\operatorname{rk}_{\mathds{Z}}E(\mathds{Q})$, and $\operatorname{sp}(m)$ for the number of primes dividing $m$ at which $E$ has split-multiplicative reduction. The zeroth Fitting ideal of an $\mathcal{R}[G]$-module $M$ is denoted by $\operatorname{Fitt}^0_{\mathcal{R}[G]}(M)$.

Mazur–Tate conjecture. The following claims are valid:

$$\theta_K^\mathrm{MT}\in I_{\mathcal{R},G}^{r+\operatorname{sp}(m)};$$

\nand

$$\theta_K^\mathrm{MT}\in\operatorname{Fitt}^0_{\mathcal{R}[G]}(S(E/K)\otimes_{\mathds{Z}}\mathcal{R}).$$

These are respectively the order-of-vanishing conjecture and the weak main conjecture of Mazur and Tate. The paper studies their $p$-parts and proves results under suitable hypotheses, but the claims are presented here as conjectures and are not resolved in general.