Liu–Xu's triangulation conjecture for special valuation spaces

Let $(X,D)$ be a klt Fano pair and let $\Delta\sim_{\mathbb{Q}}-(K_X+D)$ be a $\mathbb{Q}$-complement of $(X,D)$. Let $\mathcal{D}(X,D+\Delta)$ be the dual complex of $(X,D+\Delta)$, and let $\mathcal{D}^{KV}(X,D+\Delta)$ denote the subspace defined in the preceding question, consisting of quasi-monomial valuations up to scaling satisfying the finite-generation and klt Fano conditions there.

Liu–Xu's triangulation conjecture. There is a rational triangulation of $\mathcal{D}^{KV}(X,D+\Delta)$ such that, in the interior of $\mathcal{D}^{KV}(X,D+\Delta)$, the triangulation is locally finite. If $C^{\circ}$ is an open simplex of this triangulation, then every valuation up to scaling in $C^{\circ}$ induces a $\mathbb{G}_m$-equivariant degeneration of $X$ into a klt Fano variety, and any two valuations in $C^{\circ}$ induce isomorphic $\mathbb{G}_m$-equivariant degenerations.

This conjecture describes the global structure of the space of special valuations and asserts that valuations in a common simplex determine the same equivariant degeneration. The analogous local valuation spaces were studied previously, but the stated global triangulation and degeneration properties remain open.