Automorphic–Galois comparison conjecture for \a3-invariants

Let $\rho_{\pi,\mathfrak{p}}\colon\operatorname{Gal}(\overline{\mathbb{Q}}_p/F_{\mathfrak{p}})\to\widehat{G'}(\Omega)$ be the local Galois representation attached to $\pi$, and let $\Delta$ be the set of roots. For each $i\in\Delta$, let $\mathcal{L}_{i}(\rho_{\pi,\mathfrak{p}})\subseteq\operatorname{Hom}_{\operatorname{ct}}(F_{\mathfrak{p}}^{\times},\Omega)$ be the Galois-theoretic invariant defined from the corresponding two-dimensional quotient. Automorphic–Galois comparison conjecture. The representation $\rho_{\pi,\mathfrak{p}}$ is special and, for every $i\in\Delta$,

$$\mathcal{L}_{i}(\rho_{\pi,\mathfrak{p}})=\mathcal{L}_{i}^{(0)}(\pi,\mathfrak{p})^{{\mathbbm 1}}.$$

This is the predicted comparison between automorphic and Galois-theoretic $\mathcal{L}$-invariants; it is presented as a prediction and remains open in the stated generality.