Jacquet–Langlands invariance conjecture for automorphic L-invariants

Let $F$ be a totally real field, let ${\mathcal P}$ be a place at which both quaternion algebras split, and let $G_1$ and $G_2$ be the multiplicative groups of two quaternion algebras over $F$ that are either definite or split at a single archimedean place. Let $\Pi_1$ and $\Pi_2$ be automorphic representations of $G_1$ and $G_2$, respectively, related by the Jacquet–Langlands correspondence. For a continuous morphism $\ell:F_{\mathcal P}^\times\rightarrow {\mathbb Z}_p$, write ${\mathcal L}_{\mathcal P}(\Pi_i,\ell)$ for the associated automorphic $\mathcal L$-invariant. Jacquet–Langlands invariance conjecture. The $\mathcal L$-invariants coincide:

$${\mathcal L}_{\mathcal P}(\Pi_1,\ell)={\mathcal L}_{\mathcal P}(\Pi_2,\ell).$$

This asserts that the automorphic $\mathcal L$-invariant depends only on the corresponding representation under Jacquet–Langlands, rather than on the quaternion algebra used to realize it. The source does not specify whether the conjecture has been resolved.