Trivial zero conjecture for the cohomological and Fontaine–Mazur $\mathcal{L}$-invariants

Let $f$ be the Bianchi cuspidal eigenform and $V_p(f)$ its attached $p$-adic Galois representation. For each embedding $\sigma:F_{\mathfrak p}\hookrightarrow L$, let $\mathcal{L}^{\sigma}_{\mathfrak p}$ be the cohomological $\mathcal{L}$-invariant and let $\mathcal{L}_{\mathrm{FM}}^{\sigma}$ be the Fontaine–Mazur $\mathcal{L}$-invariant attached to $V_p(f)$. Trivial zero conjecture. For every such embedding $\sigma$, one has

$$\mathcal{L}^{\sigma}_{\mathfrak p}=\mathcal{L}_{\mathrm{FM}}^{\sigma}.$$

The equality identifies the cohomological invariant used in the paper with the $p$-adic Hodge-theoretic invariant and is needed for the subsequent rationality construction; the source supplies no resolution.