Torus-relative stability conjecture for constant scalar curvature polarised manifolds

Let $(X,L)$ be a constant scalar curvature polarised manifold. Let $R$ be its homogeneous coordinate ring, let $\chi$ be a not necessarily finitely generated filtration of $R$, and let $T\subset\operatorname{Aut}(X,L)$ be a maximal torus. Denote by $\chi_T$ the $L^2$ projection of $\chi$ along $T$.

Torus-relative stability conjecture. If

$$\lVert\chi_T\rVert_{L^2}<\lVert\chi\rVert_{L^2},$$

then

$$\operatorname{DF}(\chi)>0.$$

This proposes the analogue of the paper's theorem for constant scalar curvature polarised manifolds in the setting of arbitrary, not necessarily finitely generated, filtrations. The corresponding result is not known in this generality; the paper discusses related polynomial-filtration results and the extension to general filtrations as an open issue.