Vanishing-Lelong-number and bounded-potential conjecture for the Kähler-Ricci flow limit

In the setting of the canonical limit current conjecture, let $\varphi_\infty$ be the limiting quasi-plurisubharmonic potential.

Vanishing-Lelong-number and boundedness conjecture.

(a) $\varphi_\infty$ has vanishing Lelong number at every point.

(b) $\varphi_\infty$ is bounded.

Condition (b) implies (a), and both properties hold when $K_X$ is semiample according to the context given. Condition (a) is also equivalent to uniform exponential-integrability estimates for the smooth flow potentials; the assertions are posed as weaker properties in the general nef case.