Two-step weighted-volume degeneration conjecture for Fano fibration germs

Let $(\pi:X\rightarrow Y,p)$ be a Fano fibration germ. Let $\operatorname{Val}_{\pi,p}^*$ be the relevant space of valuations and let $\mathbb W$ denote the weighted volume. For a valuation $v_*$, write $\operatorname{Gr}_{v_*}$ for the associated graded algebra, and form the schemes and morphisms specified below.

Two-step degeneration conjecture for weighted volume.

1. There is a unique minimizer $v_*$ of $\mathbb W$ on $\operatorname{Val}_{\pi,p}^*$, which is quasi-monomial.
2. The graded ring $\operatorname{Gr}_{v_*}(\mathcal O_{Y,p})$ is finitely generated and $\bigoplus_{m\geq 0}\operatorname{Gr}_{v_*}(R_m)$ is a finitely generated algebra over $\operatorname{Gr}_{v_*}(\mathcal O_{Y,p})$.
3. Let

$$W=\operatorname{Spec}(\operatorname{Gr}_{v_*}(\mathcal O_{Y,p})),\qquad Z=\operatorname{Proj}_{W}\left(\bigoplus_{m\geq 0}\operatorname{Gr}_{v_*}(R_m)\right).$$

Then $v_*$ induces a vector field $\xi$ on $Z$ which descends to $W$. The natural map $\widetilde\pi:Z\rightarrow W$ and $\xi$ define a K-semistable polarized Fano fibration. Moreover, $v_*$ is uniquely characterized by this K-semistability property. 4. There is an equivariant test configuration degenerating $(\widetilde\pi:Z\rightarrow W,\xi)$ to a unique K-polystable polarized Fano fibration $(\widehat\pi:\mathcal S\rightarrow\mathcal C,\xi)$.

The conjecture proposes a canonical algebraic two-step degeneration of a Fano fibration germ, first through weighted-volume minimization and then through a test configuration to a K-polystable object. Its existence, finite generation, uniqueness, and compatibility assertions are open in the stated generality.