Mazur–Tate's leading-term conjecture for elliptic curves

Let $E$ be an elliptic curve, let $L$ be an abelian number field with Galois group $G_L=\operatorname{Gal}(L/\mathds{Q})$, and let $m'$ be its conductor. Assume that $m'$ is a product of primes at which $E$ has split-multiplicative reduction. For each such prime $\ell$, let $q_{E,\ell}\in\mathds{Q}_\ell^\times$ be the Tate multiplicative period, let $\operatorname{Tam}_\ell=\operatorname{ord}_\ell(q_{E,\ell})$, and let $\operatorname{rec}_\ell: \mathds{Q}_\ell^\times\to\mathcal{D}^{(\ell)}_L\subseteq G_L$ be the local reciprocity map. Let $I_{\mathcal{R},G_L}$ be the augmentation ideal and $\operatorname{sp}(m')$ the number of prime divisors of $m'$.

Mazur–Tate leading-term conjecture. One has

$$\theta_L^\mathrm{MT}\equiv\theta_{\mathds{Q}}^\mathrm{MT}\cdot\prod_{\ell\mid m'}\left(\operatorname{Tam}_\ell^{-1}\cdot(\operatorname{rec}_\ell(q_{E,\ell})-1)\right)\pmod{I_{\mathcal{R},G_L}^{\operatorname{sp}(m')+1}}.$$

This predicts the leading augmentation-ideal term of the Mazur–Tate element after adjoining split-multiplicative primes. The source presents it as a conjecture of Mazur and Tate; no general resolution is stated.