The extremal K-polystability conjecture for smooth Kähler test configurations

Let $(M,J,2\pi c_1(L))$ be the polarized manifold in the setting of Theorem~, and let $\check{\mathbb T}$ be the relevant torus. A $\check{\mathbb T}$-equivariant smooth Kähler test configuration is a smooth Kähler test configuration $({\mathscr M},{\mathscr A})$ associated to $(M,J,2\pi c_1(L))$ with the stated $\check{\mathbb T}$-equivariance; it has a reduced central fibre when the fibre over $0$ is reduced. The configuration is a product when it is the product test configuration.

Extremal K-polystability conjecture. In the setting of Theorem~, for any $\check{\mathbb T}$-equivariant smooth Kähler test configuration $({\mathscr M},{\mathscr A})$ associated to $(M,J,2\pi c_1(L))$, which has a reduced central fibre and is not a product, we have

$${\mathscr F}^{\mathrm{ext}}_K({\mathscr M},{\mathscr A})>0.$$

This strengthens the nonnegativity required for extremal Sasaki structures to positivity for every non-product smooth Kähler test configuration with reduced central fibre, and is intended as the polystability statement in this setting. The supplied text gives no resolution of the conjecture.