Duality for boundary pro-étale cohomology

Let $Y$ be a smooth partially proper rigid analytic variety over $K$ of dimension $d$, let $\partial Y$ denote its boundary, and let ${\mathbb R}_{\operatorname{pro\acute{e}t}}(\partial Y_C,{\bf Q}_p)$ be the corresponding TVS-valued pro-étale cohomology complex. Boundary pro-étale duality conjecture. There is a natural quasi-isomorphism in an as-yet unspecified category $\mathcal D(?)$,

$${\mathbb R}_{\operatorname{pro\acute{e}t}}(\partial Y_C,{\bf Q}_p)\stackrel{\sim}{\to}{\rm R}{\cal Hom}_{?}({\mathbb R}_{\operatorname{pro\acute{e}t}}(\partial Y_C,{\bf Q}_p(d))[2d-1],{\bf Q}_p).$$

This is explicitly presented as a desired extension of geometric duality to the boundary; the ambient category and the appropriate internal Hom are not known, so the problem remains open.