Let $X$ be a compact Polish space and $K(X)$ the hyperspace of closed subspaces of $X$ with the Vietoris/Hausdorff metric topology.

Question: If $A$ is an analytic subset of $X$, what is the complexity of the set $\{C\in K(X): C\subseteq A\}$?

This set is clearly $\mathbf{\Pi}^1_2$, but can this be reduced to $\mathbf{\Delta}^1_2$ or even to (co-)analytic?

notcoincide with the closed subspaces of $X$ which happen to be contained in $A$... $\endgroup$