Greenberg–Benois exceptional zero conjecture for $\mathrm{GL}(3)$

Let $L_p^{(c)}(\pi,s)=\mathscr{L}_p^{(c)}(\pi,\langle-\rangle^s)$ for $s\in\mathbb{Z}_p$, where $\langle-\rangle:\mathbb{Z}_p^{\times}\to1+2p\mathbb{Z}_p$ is the natural map. Let $e_{\infty}^{(c)}$, $E_p^{(c)}$, $\mathcal{L}^{\operatorname{FM}}_{\pi,c+1}$, $L(\pi,c)$, and $\Omega_{\pi,c}$ have the meanings specified in the preceding setup. Exceptional zero conjecture.

$$\left.\frac{d}{ds}L_p^{(c)}(\pi,s)\right|_{s=0}=e_{\infty}^{(c)}(\pi_\infty,c)\,E_p^{(c)}(c)\,\mathcal{L}^{\operatorname{FM}}_{\pi,c+1}\,\frac{L(\pi,c)}{\Omega_{\pi,c}}.$$

The formula predicts that the first derivative at the trivial character is governed by the Fontaine–Mazur $\mathcal{L}$-invariant. The conjecture is conditional on the existence of the interpolating $p$-adic $L$-function, and no resolution is given in the supplied text.