Chida–Mok–Park's Jacquet–Langlands conjecture for Hilbert modular $L$-invariants

Fix a prime number $p$. Let $F$ be a totally real field, set $g=[F:{\mathbb Q}]$, and let ${\mathfrak p}$ be a prime ideal of $F$ above $p$. Let $f_\infty$ be a Hilbert eigen newform with even weight $(k_1,\ldots,k_g,w)$ and level divisible exactly by ${\mathfrak p}$, and let $\mathbf f$ be an automorphic form on a totally definite quaternion algebra over $F$ of the same weight. Assume that $\mathbf f$ is new at ${\mathfrak p}$ and satisfies

$$U_{\mathfrak p}\mathbf f={\mathcal N}{\mathfrak p}^{w/2}\mathbf f.$$

The associated Fontaine–Mazur and Teitelbaum-type invariants are denoted by ${\mathcal L}_{FM}(f_\infty)$ and ${\mathcal L}_T(\mathbf f)$, respectively. Chida–Mok–Park's conjecture. If $f_\infty$ and $\mathbf f$ are associated to each other by the Jacquet–Langlands correspondence, then

$${\mathcal L}_{FM}(f_\infty)={\mathcal L}_T(\mathbf f).$$

This is the precise Hilbert modular analogue of the equality between Fontaine–Mazur and Teitelbaum-type $L$-invariants. The supplied source does not give evidence that the conjecture has been resolved.