Let $K$ be a $p$-adic field with residue field $k$, and $X$ a variety over $k$. The rigid cohomology of $X$ over $k$ can be described very neatly using Grosse--Kloenne's notion of dagger spaces: embed $X$ into the special fibre of a smooth proper formal scheme $P$ over $\mathcal{O}_K$; then the tube of $X$ in $P$ naturally has the structure of a dagger space $\mathcal{X}^\dagger$, and the de Rham cohomology of $\mathcal{X}^\dagger$ is the rigid cohomology of $X$.
Here's my question: how should one interpret the compactly-supported de Rham cohomology of $\mathcal{X}^\dagger$?
This doesn't seem to be computing compactly-supported rigid cohomology in general: I believe this is true if $X$ is open in $P_k$, but if $X$ is closed in $P_k$ it looks more like rigid cohomology of $P_k$ with support in $X$. Is there any literature where these sorts of cohomology groups come up?
(If we had Poincare duality for $\mathcal{X}^\dagger$, then that would help to answer the question, but I can't find any references for Poincare duality for dagger spaces that aren't either affinoid, proper, or Stein.)
