Duality for Arithmetic $p$-adic Pro-\'etale Cohomology of Analytic Spaces

Let $K$ be a finite extension of $\mathbb{Q}_p$. We prove that the arithmetic $p$-adic pro-\'etale cohomology of smooth partially proper spaces over $K$ satisfies a duality, as conjectured by Colmez, Gilles and Nizio{\l}. We derive it from the geometric duality on the Fargues-Fontaine curve by Galois descent techniques of Fontaine.