K-stable but not uniformly K-stable smooth polarized varieties

Construct a normal polarized variety that is K-stable but not uniformly K-stable, preferably within the smooth or klt category rather than relying on strictly log canonical or deminormal examples.

Background

The paper distinguishes ordinary K-polystability from uniform relative K-polystability and constructs a smooth polarized fivefold whose reduced Donaldson–Futaki quotient has infimum zero. This shows failure of a positive uniform margin, but the resulting variety has a nontrivial connected automorphism group and is not presented as a resolution of the stronger K-stability-versus-uniform-K-stability question.

The cited conjecture concerns the existence of K-stable, non-uniformly K-stable normal polarized varieties. The paper notes that known examples at the time rely on strictly log canonical singularities or deminormality and that extending the phenomenon to smooth or klt varieties is difficult.