The minimizer conjecture for the pluripotential stability threshold

Let $(X,L)$ be a polarized manifold, and let $\delta$ and $\delta_{\mathrm{pp}}$ denote the stability threshold and pluripotential-theoretic stability threshold, respectively. A pluricomplex Green function is the function $G$ associated with the limiting singularities of Aubin's continuity path in the Fano setting. Minimizer conjecture. If $\delta_{\mathrm{pp}}\leq 1$, then $\delta_{\mathrm{pp}}$ has a minimizer. If $X$ is Fano, $L=-K_X$, and $\delta<1$, then the pluricomplex Green function $G$ in the sense of McCleerey–Collins–Tian is a minimizer of $\delta_{\mathrm{pp}}$. The conjecture seeks explicit minimizers for the pluripotential stability threshold; the source does not state a resolution.