Zhang's uniform lower-bound conjecture for the Kähler-Ricci potential

Let $X$ be a compact Kähler manifold, let $\varphi(t)$ solve the parabolic complex Monge–Ampère equation associated with the Kähler-Ricci flow, and suppose the solution exists on $X\times[0,T)$ with $T<\infty$. Zhang's uniform lower-bound conjecture. There is a constant $C>0$ such that

$$\varphi(t)\geqslant -C$$

on $X\times[0,T)$. The source identifies this as a conjecture explicitly raised by Zhang; it records partial progress but no resolution in general.