Finite-slope p-adic L-functions for ${\mathrm{G}\mathrm{L}}_3$

Let $\mathrm{\Pi }$ be a regular algebraic cuspidal automorphic representation of ${\mathrm{G}\mathrm{L}}_3({\mathbb{A}}_{\mathbb{Q}})$ of weight $(a,0,-a)$, and let $p$ be a prime. Generalising work of Loeffler–Williams in the nearly-ordinary case, we construct $p$-adic $L$-functions for each small slope regular $P_1$-refinement of $\mathrm{\Pi }$ at $p$ consistent with the conjectures of Coates–Perrin-Riou and Panchishkin.