Jacquet–Langlands equality of localized new homology

Let $Y(K_{\Sigma})$ and $Y'(K_{\Sigma})$ be a Jacquet–Langlands pair, let $\mathbf{T}$ be the relevant Hecke algebra, and let $\mathfrak{m}$ be a non-Eisenstein maximal ideal. Write $H_1(Y(K_{\Sigma}),\mathbf{Z})^{\mathrm{new}}_{\mathfrak{m}}$ and $H_1(Y'(K_{\Sigma}),\mathbf{Z})^{\mathrm{new}}_{\mathfrak{m}}$ for the localized new homology groups. Jacquet–Langlands homological conjecture. If $\mathfrak{m}$ is not Eisenstein, then

$$\left|H_1(Y(K_{\Sigma}),\mathbf{Z})^{\mathrm{new}}_{\mathfrak{m}}\right|=\left|H_1(Y'(K_{\Sigma}),\mathbf{Z})^{\mathrm{new}}_{\mathfrak{m}}\right|.$$

This predicts equality of the localized torsion orders on the two sides of a Jacquet–Langlands correspondence; the source explains it heuristically through a common Hecke algebra.