The $n=3$ local identification conjecture for $\pi^\alpha(\rho_p)$

Let $\rho_p:{\rm Gal}(\overline{\mathbb Q}_p/{\mathbb Q}_p)\to{\rm GL}_3(E)$ be semistable as in the preceding setup, with $\dim_E W=3$ and $N^2\ne0$. The $n=3$ local identification conjecture. One has

$$\pi^\alpha(\rho_p)\simeq[\![({\mathcal F}_\alpha\circ{\mathcal E}_\alpha)^{-1}(\operatorname{Fil}^{\rm \max}_\alpha(\rho_p))]\!].$$

This conjecture says that the representation attached directly to $\rho_p$ agrees with the representation obtained from the maximal filtration line; the surrounding text records partial results for dimension three but no proof of this final identification.