Higher Theta-stratification conjecture for equivariant Fano fibrations

Consider $T$-equivariant Fano fibrations $\pi\colon X\to Y$, fixing the multi-Hilbert function of $T\curvearrowright Y$ and the Hilbert polynomial of $-K_X$ on the fibers of $\pi$. Higher Theta-stratification conjecture. There is a sufficiently large parameter scheme $H\curvearrowleft G$ of $T$-equivariant Fano fibrations, with $G$ preserving isomorphism classes, such that the weighted volume $\mathbb{W}(-)$ is lower semicontinuous and induces a higher $\Theta$-stratification with finitely many strata $\{\mathcal{Z}_c:=\{\mathbb{W}(-)=c\}\}_c$ on $[H/G]$, encoding the generalized test configurations familywise as $\mathcal{Z}_c\times[U_\sigma/T]\to\mathcal{Z}_c$. This is proposed as a moduli-theoretic structure extending the parameter-space construction and incorporating generalized test configurations; the source gives no resolution.