The compactification conjecture for constructible sheaf data on Alexandrov spaces

Fix a finite field $\mathbb{F}$ and let $\mathcal{L}_{\mathbb{F}}(n,D,\kappa)$ consist of pairs $(X,[\mathcal{F}])$, where $X$ is an Alexandrov space of Hausdorff dimension at most $n$, diameter at most $D$, curvature at least $\kappa$, and $\mathcal{F}$ is a constructible object of $D^b(\operatorname{Sh}_{\mathbb{F}}(X))$. The sheaf-moduli topology conjecture. There should exist a Hausdorff topology on this set satisfying the stated sequential compatibility with Gromov–Hausdorff convergence and stabilization of pushforward objects, continuity of the map to constructible-function data and to Alexandrov spaces, closedness under enlargement of $(n,D,\kappa)$, compactness of the closure of smooth manifolds with rank-one constant sheaf, and finiteness of the fibers of the map from that closure to the Alexandrov-space moduli. The source gives no proof of these properties; they are proposed as a framework for compactifying sheaf-theoretic data.