Ogg's conjecture on the Shimura subgroup of J_0(p)

Let $p$ be prime. Let $\mathcal{S}(p)$ be the Shimura subgroup of $J_0(p)$, and let $\mathcal{M}(p)$ be the largest $\mu$-type subgroup of $J_0(p)$. Ogg's Shimura-subgroup conjecture.

$$\mathcal{S}(p)=\mathcal{M}(p).$$

This is the dual counterpart of Ogg's cuspidal torsion conjecture. The source says that Mazur proved it using properties of the Eisenstein ideal and the Gorenstein property of the relevant Hecke algebras.