Algebraic proof linking non-Archimedean stability notions

Develop an algebraic proof linking widehat{K}-polystability and, when the automorphism group is discrete, uniform K^ beta-stability to the other equivalent K-stability notions listed in Corollary B, including Aut^0(X,L)-uniform K-stability, uniform K-polystability, and the corresponding stability notions for models.

Background

Corollary B states equivalences among several stability conditions for a smooth polarized variety (X,L), including classical equivariant uniform K-stability, uniform K-polystability, stability for log-discrepancy models and models, uniform and non-uniform \widehat{K}-polystability, and, when the automorphism group is trivial, uniform K^\beta-stability. The paper explains that some of these equivalences are obtained algebraically, while others follow from the main theorem and results cited from BoucksomJonsson and DarvasZhang.

The unresolved issue is whether the remaining implications can also be established by a direct algebraic argument, rather than through the analytic and non-Archimedean methods used in the paper and the cited works. In particular, the passage explicitly identifies the absence of an available algebraic proof connecting \widehat{K}-polystability (and, in the discrete-automorphism case, uniform K^\beta-stability) with the other stability notions.