Higher $mathcal{L}$-invariants for $mathrm{GL}_3(mathbb{Q}_p)$ and local-global compatibility

Let $rho_p$ be a $3$-dimensional $p$-adic semi-stable representation of $mathrm{Gal}(overline{mathbb{Q}_p}/mathbb{Q}_p)$ with Hodge-Tate weights $(0,1,2)$ (up to shift) and such that $N^2 e 0$ on $D_{mathrm{st}}(rho_p)$. When $rho_p$ comes from an automorphic representation $pi$ of $G(mathbb{A}_{F^+})$ (for a unitary group $G$ over a totally real field $F^+$ which is compact at infinite places and $mathrm{GL}_3$ at $p$-adic places), we show under mild genericity assumptions that the associated Hecke-isotypic subspaces of the Banach spaces of $p$-adic automorphic forms on $G(mathbb{A}_{F^+}^infty)$ of arbitrary fixed tame level contain (copies of) a unique admissible finite length locally analytic representation of $mathrm{GL}_3(mathbb{Q}_p)$ which only depends on and completely determines $rho_p$.