Endomorphism ring of $J_0(p)$ and Hecke operators

Let $$J_0(p)$$ be Jacobian of the modular curve $$X_0(p)$$ over $$\mathbb Q$$ where $$p$$ is a prime, consider the subring $$\mathbb T$$ inside $$End_{\mathbb Q}(J_0(p))$$ generated by Hecke operators $$T_n$$ for all $$(n, p)=1$$. Then what are $$\mathbb T$$ and $$End_{\mathbb Q}(J_0(p))$$ as $$\mathbb Z$$-algebras? Do we have $$\mathbb T=End_{\mathbb Q}(J_0(p))$$? If not, what is the index /difference?

If we tensor both sides with $$\mathbb Q$$, then they are equal by the work of Ribet, and the ring structure is given by products of totally real fields which are precisely coefficient fields of weight $$2$$ level $$p$$ cusp forms. Here I am more interested in the integral structures.

Mazur proved that $$\mathbb{T}' = \text{End}_{\mathbf{Q}}(J_0(p))$$ where $$\mathbb{T}'$$ is generated by $$\textit{all}$$ the Hecke operators $$T_n$$ for $$n\geq 1$$ (including $$n$$ divisible by $$p$$). Note that $$\mathbb{T}' = \mathbb{T}[U_p]=\mathbb{T}[w_p]$$ where $$U_p$$ is the $$p$$th Hecke operator and $$w_p = -U_p$$ is the Atkin--Lehner involution (cf. Mazur's Eisenstein ideal paper, Proposition II.9.5 p. 95). So the question becomes whether $$w_p$$ (or $$U_p$$) belongs to $$\mathbb{T}$$. This is a local problem at maximal ideal of residue characteristic $$2$$. At Eisenstein maximal ideal this is known to be true, but I'm not quite sure at non-Eisenstein ideal.