Strong properness conjecture for K-energy on Sasaki manifolds

Let $(M,g)$ be a $(2n+1)$-dimensional Sasaki manifold with transverse Kähler form

$$\omega_g^T=\frac{\pi}{n+1}c_1^B(M).$$

Suppose that $M$ admits a transverse Sasaki–Einstein metric. Let $K$ be a maximal compact subgroup of ${\rm Aut}^T(M)$, and let $I(\cdot)$ be the functional defined in the source. For a transverse Kähler potential $\psi$ of $\omega_g^T$, write $\psi_\tau$ for the induced transverse Kähler potential under $\tau\in Z({\rm Aut}^T(M))$. Strong properness conjecture. There exist $\delta,C_\delta>0$ such that, for every $K$-invariant transverse Kähler potential $\psi$ of $\omega_g^T$,

$$\mathcal K(\psi)\geq \delta\inf_{\tau\in Z({\rm Aut}^T(M))}I(\psi_\tau)-C_\delta.$$

This conjecture extends the stated strong properness result from $G$-Sasaki manifolds with the required symmetry to general Sasaki manifolds admitting a transverse Sasaki–Einstein metric. It asserts properness of K-energy modulo the central transverse holomorphic automorphisms; the supplied text gives no resolution, so its status remains open.