Integral-closure conjecture for the plus and minus Hecke algebras

Let $p$ be an odd prime, let $k$ be a positive integer, and let $\Gamma=\Gamma_0(p)$. Let $\mathbf{T}^{\operatorname{new}}$ act on the new subspace of cusp forms, and let $w_p$ be the Fricke involution. Define

$$\mathbf{T}^{+}=\mathbf{T}^{\operatorname{new}}/(U_p+p^{(k-2)/2}),\qquad \mathbf{T}^{-}=\mathbf{T}^{\operatorname{new}}/(U_p-p^{(k-2)/2}),$$

where these algebras act on the $+1$ and $-1$ eigenspaces, respectively. Let $\overline{\mathbf{Z}}_p$ denote the algebraic integers in an algebraic closure of $\mathbf{Q}_p$.

Integral-closure conjecture. If $k<p-1$, then $\mathbf{T}^{+}$ and $\mathbf{T}^{-}$ are integrally closed. Equivalently, every congruence between distinct eigenforms in $S_k(\overline{\mathbf{Z}}_p)$ occurs between eigenforms with opposite $w_p$-eigenvalues.

For $k=2$, the paper explains that opposite-sign congruences cannot occur because $p$ is odd, so this recovers the assertion that $p$ does not divide the normalization index. The general assertion is presented as a conjecture about the source of congruences.