Local closedness and finiteness conjectures for Kollár valuations

Let $x\text{?}$ denote the klt singularity and let $D$ be a $\mathbb{Q}$-complement such that {x\} is the only lc center of $(X,\Delta+D)$. Write $\mathcal{D}(X,\Delta+D)$ for the associated dual complex and $\mathcal{D}^{\rm KV}(X,\Delta+D)$ for its locus of Kollár valuations. A rational triangulation is a triangulation whose vertices have rational coordinates, and a finite triangulation has finitely many simplices. A cell is realized in a Kollár model if it is represented by Kollár valuations on such a model.

Local closedness and finiteness conjectures. The locus $\mathcal{D}^{\rm KV}(X,\Delta+D)$ satisfies all of the following:

1. There is a rational triangulation of $\mathcal{D}(X,\Delta+D)$ such that for each open simplex $C^{\circ}$, $\mathcal{D}^{\rm KV}(X,\Delta+D)\cap C^{\circ}$ is open in its closure $\overline{\mathcal{D}^{\rm KV}(X,\Delta+D)\cap C^{\circ}}\subset C^{\circ}$.
2. There is a rational triangulation of $\mathcal{D}(X,\Delta+D)$ such that for each open simplex $C^{\circ}$, $\mathcal{D}^{\rm KV}(X,\Delta+D)\cap C^{\circ}\subset C^{\circ}$ is open.
3. If $\mathcal{D}(X,\Delta+D)=\mathcal{D}^{\rm KV}(X,\Delta+D)$, then there is a finite triangulation of $\mathcal{D}(X,\Delta+D)$ such that each simplex can be realized on a Kollár model.
4. If $C\subseteq\mathcal{D}^{\rm KV}(X,\Delta+D)$ is a closed cell of $\mathcal{D}(X,\Delta+D)$ after a triangulation given by a log resolution, then there is a finite triangulation $C=\bigcup_{i=1}^{N}C_i$ such that each $C_i$ is realized in a Kollár model.

These conjectures refine the known nonemptiness, local structure, and path connectedness results for the locus of Kollár valuations. The source does not provide evidence resolving any of the four assertions.