The bounded-potential conjecture for nef classes on compact Kähler manifolds

Let $X$ be a compact Kähler manifold and let $[\alpha]$ be a nef $(1,1)$ class such that $[\alpha]+\lambda c_1(X)$ is a Kähler class for some $\lambda>0$. A closed positive current in $[\alpha]$ is said to have minimal singularities when its potential is no more singular than the potentials of all other closed positive currents in the class. Bounded-potential conjecture. A closed positive current with minimal singularities in the class $[\alpha]$ has bounded potential. The source says this conjecture is not about the Kähler-Ricci flow and gives no general resolution.