Arithmetic Poincaré duality for pro-étale cohomology of Stein dagger varieties

Let $K$ be the base field, and let $X$ be a smooth Stein dagger variety over $K$, geometrically irreducible and of dimension $d$. Write $H^i_{\operatorname{pro\acute{e}t}}(X,\mathbf{Q}_p(j))$ and $H^i_{\operatorname{pro\acute{e}t},c}(X,\mathbf{Q}_p(j))$ for pro-étale cohomology with and without compact support, and let $\mathbb D(-)={\mathrm R}\underline{\operatorname{Hom}}_{\mathbf{Q}_p}(-,\mathbf{Q}_p)$. Arithmetic Poincaré duality conjecture. The groups $H^i_{\operatorname{pro\acute{e}t}}(X,\mathbf{Q}_p(j))$ and $H^i_{\operatorname{pro\acute{e}t},c}(X,\mathbf{Q}_p(j))$ are nuclear Fréchet and of compact type, respectively, and there are (quasi-)isomorphisms in $\mathcal{D}(\mathbf{Q}_{p,\Box})$:

$${\mathrm R}\Gamma_{\operatorname{pro\acute{e}t}}(X,\mathbf{Q}_p(j))\simeq \mathbb D\bigl({\mathrm R}\Gamma_{\operatorname{pro\acute{e}t},c}(X,\mathbf{Q}_p(d+1-j))[2d+2]\bigr),$$ $$H^i_{\operatorname{pro\acute{e}t}}(X,\mathbf{Q}_p(j))\simeq H^{2d+2-i}_{\operatorname{pro\acute{e}t},c}(X,\mathbf{Q}_p(d+1-j))^*,$$ $$H^i_{\operatorname{pro\acute{e}t},c}(X,\mathbf{Q}_p(j))\simeq H^{2d+2-i}_{\operatorname{pro\acute{e}t}}(X,\mathbf{Q}_p(d+1-j))^*.$$

This is proposed as an arithmetic Poincaré duality extending the dimension-one duality established earlier in the paper to smooth Stein dagger varieties of arbitrary dimension. The status of the functional-analytic assertions and the duality in this generality is not resolved in the supplied text.