General existence conjecture for higher Theta-stratifications of Fano-fibration moduli

Let $\mathcal{M}^o$ be a finite-type algebraic $\mathbbm{k}$-stack underlying a $\mathbb{Q}$-Gorenstein faithfully flat affine family of $\mathbb{Q}$-Fano fibrations, equipped with a section of the base morphism. General higher $\Theta$-stratification conjecture. There is a monomorphism $\mathcal{M}^o\to\mathcal{M}$ into another finite-type quotient algebraic $\mathbbm{k}$-stack carrying an extending $\mathbb{Q}$-Gorenstein family of $\mathbb{Q}$-Fano fibrations and a higher $\Theta$-stratification defined by the lower semicontinuous, constructible weighted-volume function, with finitely many strata $\{\mathcal{Z}_c:=\{\mathbb{W}(-)=c\}\}_{c\in\mathbb{R}}$ encoding the generalized test configurations familywise as $\mathcal{Z}_c\times[U_\sigma/T]\to\mathcal{Z}_c$. This is a proposed generalization of the preceding parameter-space conjecture toward moduli and properness; the source gives no resolution.