Kähler–Einstein existence and relative Kähler–Einstein potentials for Fano manifolds

Let $(X,\omega)$ be a Fano manifold. Write $\operatorname{Aut}(X)^{\circ}$ for the identity component of its automorphism group, let $\operatorname{Id}$ denote the identity automorphism, and let $\mathcal{M}_{KE}$ and $\mathcal{M}_{klt}^{+}$ denote the spaces of Kähler–Einstein potentials and klt positive model potentials, respectively.

Kähler–Einstein relative singularity conjecture. If

$$\operatorname{Aut}(X)^{\circ}=\{\operatorname{Id}\},$$

then

$$0\in\mathcal{M}_{KE}\Longleftrightarrow\mathcal{M}_{KE}=\mathcal{M}_{klt}^{+}.$$

This conjecture proposes that, for Fano manifolds with discrete automorphism group, the existence of a Kähler–Einstein metric is equivalent to every klt positive model potential being Kähler–Einstein. The source presents it as suggested by the sharpness example for the preceding alpha-invariant criterion; no resolution is stated.