Duality for $p$-adic geometric pro-étale cohomology

We prove that $p$-adic geometric pro-étale cohomology of smooth partially proper rigid analytic varieties over $p$-adic fields seen in the category of Topological Vector Spaces satisfies a Poincaré duality as we have conjectured. This duality descends, via fully-faithfulness results of Colmez-Nizioł, from a Poincaré duality for solid quasi-coherent sheaves on the Fargues-Fontaine curve representing this cohomology. The latter duality is proved by passing, via comparison theorems, to analogous sheaves representing syntomic cohomology and then reducing to Poincaré duality for ${\mathbf{B}}_{\mathrm{s}\mathrm{t}}^{+}$-twisted Hyodo-Kato and filtered ${\mathbf{B}}_{\mathrm{d}\mathrm{r}}^{+}$-cohomologies that, in turn, reduce to Serre duality for smooth Stein varieties – a classical result.