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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 $N^2\ne 0$ 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 $\rho_p$.

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