Local dependence of the mod p Jacquet–Langlands correspondence

Let $\mathfrak{m}$ be a maximal ideal of the Hecke algebra, with associated Galois representation $\bar{\rho}$. Let $\mathbf{F}[\mathfrak{m}]$ denote the corresponding Hecke-isotypic representation, and let $\pi^p(\bar{\rho})$ denote its prime-to-$p$ automorphic factor. Mod $p$ Jacquet–Langlands conjecture. There is a $D^{\times}(\mathbb{Q})$-equivariant isomorphism

$$\mathbf{F}[\mathfrak{m}] \simeq \sigma \otimes \pi^p(\bar{\rho}),$$

where $\sigma$ is a $D^{\times}(\mathbb{Q}_p)$-representation depending only on the restriction $\bar{\rho}_p$. The conjecture formulates the expected factorization into a local representation at $p$ and the prime-to-$p$ part, with the local factor determined solely by local Galois data.