K-polystability and reduced uniform K-stability of log Fano cone singularities

We prove that a log Fano cone $(X,\mathrm{\Delta },{\xi }_0)$ satisfying ${\delta }_{\mathbb{T}}(X,\mathrm{\Delta },{\xi }_0)\ge 1$ is K-polystable for normal test configurations if and only if it is K-polystable for special test configurations. We also establish the reduced uniform K-stability of $(X,\mathrm{\Delta },{\xi }_0)$ and show that it is equivalent to K-polystability.