The Loewy filtration destabilisation conjecture for polarised varieties

Let $(X,L)$ be a polarised variety, and let the Loewy filtration be the canonical $\operatorname{Aut}(X,L)$-equivariant filtration of its co-ordinate ring, which is trivial if and only if $\operatorname{Aut}(X,L)$ is reductive.

Loewy filtration destabilisation conjecture. If $(X,L)$ has non-reductive automorphism group, then the Loewy filtration destabilises $(X,L)$.

This conjecture predicts that the canonical filtration associated with a non-reductive automorphism group has strictly negative Donaldson–Futaki invariant, and therefore obstructs K-polystability. It is motivated by the expectation that a smooth polarised variety admitting a cscK metric should have reductive automorphism group, while the conjecture addresses the corresponding filtration-theoretic destabilisation.