Codogni–Dervan's Loewy filtration conjecture for Fano manifolds

Let $X$ be a Fano manifold with automorphism group $\operatorname{Aut}(X)$. A Loewy filtration is the canonical filtration of the section ring constructed from the action of $\operatorname{Aut}(X)$. Assume that $\operatorname{Aut}(X)$ is non-reductive and that the Loewy filtration of $X$ is finitely generated. The filtration then induces a test configuration $(\mathcal {X}_{\mathrm{Loe}},\mathcal {L}_{\mathrm{Loe}})$ of $X$. Codogni–Dervan's Loewy filtration conjecture. The induced test configuration $(\mathcal {X}_{\mathrm{Loe}},\mathcal {L}_{\mathrm{Loe}})$ destabilizes $X$. This is a special case of Codogni and Dervan's conjecture, restricted to the case where the Loewy filtration is finitely generated. The paper's abstract states that it gives a counterexample to their conjecture, so this claim is refuted.