Sun–Zhang's boundedness conjecture for equivariant Fano fibrations

Let $T$ be a torus and let $c>0$ be a real number. Consider isomorphism classes of $T$-equivariant $(\mathbb{Q})$-Fano fibrations $T\curvearrowright(X\xrightarrow{\pi}Y)$, with weighted volume $\mathbb{W}(\pi)$. Sun–Zhang's boundedness conjecture. If $\mathbb{W}(\pi)\ge c$, then these isomorphism classes form a bounded family, parametrized inside a finite-type, quasi-compact $\mathbbm{k}$-scheme. The conjecture is intended to provide boundedness needed for moduli constructions of K-semistable equivariant Fano fibrations; the source gives no resolution.