Let $h^{ord}(N,\mathcal{O})$ be the $p$-ordinary Hecke algebra, and $\mathfrak{m}$ be a maximal ideal of the semi local ring $h^{ord}(N,\mathcal{O})$ corresponding to a residual representation $\bar{\rho}$ which is $p$-distinguish and minimal and such that the restriction of $\bar{\rho}$ to $G_{\mathbb{Q}(\sqrt{(-1)^{(p-1)/2}p)}}$ is absolutely irreducible.

let $T$ be the localisation of $h^{ord}(N,\mathcal{O})$ by $\mathfrak{m}$.

In the literature I see that we assume often that the tame level is a square free (since any semi-stable elliptic curves has a square free conductor)

There is some modularity lifting theorems $R^{ord}=T$ without the assumption that the tame level of $T$ is a square free?

I don't see immediately when we use this condition in the Horizontal control (in the Hecke side)?