Higher $\Theta$-stratification by $h$-invariant of Fano varieties

Let $S$ be a finite type $\mathbbm{k}$-scheme and let $\pi\colon\mathcal{X}\to S$ be a $\mathbb{Q}$-Gorenstein family whose $h$-invariants and volumes are constant on closed fibers. Let $N$ be a finite-rank free abelian group, let $\tau\subset N\otimes\mathbb{R}$ be a rational polyhedral cone, and let $U_\tau$ be the affine toric variety corresponding to $N_{\mathbb{R}}\supset\tau$; write $\Delta_\xi$ for the $N$-spectrum in $\S 2$. For positive integer $n$, positive real number $V$, and real number $h$, let $\mathcal{M}_{n,V,h^+}$ be the finite-type moduli Artin stack of $n$-dimensional $\mathbb{Q}$-Fano varieties with anticanonical volume $(-K_X)^n=V$ and $h(X)\geq h$, with finite stratification $\{\mathcal{M}_{n,V,h'}\}_{h'\geq h}$ by the values of $h(X)$. Higher $\Theta$-stratification conjecture. The $h$-invariants of $\mathbb{Q}$-Fano families and their corresponding first-step degenerations satisfy the following properties: (1) there exist $N$, $\tau$, and $\xi\in S$ and a $\mathbb{Q}$-Gorenstein family $\tilde{\pi}\colon\tilde{\mathcal{X}}\to U_\tau\times S$ restricting to $\pi$ and to the corresponding first-step degeneration of each $X_s$ on the specified loci; and (2) $\mathcal{M}_{n,V,h^+}$ and its natural finite stratification admit an étale-locally liftable higher $\Theta$-stratification encoding these first-step degenerations. This predicts a higher-\ $\Theta$-stratified structure on bounded moduli of $\mathbb{Q}$-Fano varieties compatible with first-step degenerations, extending properness methods for K-moduli of shrinking Kähler–Ricci solitons. The source presents these properties as expected analogues of existing results, and gives no resolution of the conjecture.