Zero-Lelong-number conjecture for degenerate complex Monge–Ampère flows

Let $\Omega\subset\mathbb{C}^n$ be the strictly pseudoconvex domain and let $u_0$ be the initial datum considered in Theorem. Replace the assumption

$$u_0\in L^{\infty}(\tilde{\Omega})$$

by the condition that $u_0$ has zero Lelong numbers. Then zero-Lelong-number conjecture. there exists a unique function $u\in C^{\infty}(\bar{\Omega}\times (0,T))$ satisfying the plurisubharmonicity, parabolic complex Monge–Ampère, and boundary conditions in the paper, such that

$$u(\cdot,t)\longrightarrow u_0\quad\text{in }L^1(\Omega).$$

This asks whether the existence and uniqueness result proved for bounded initial data extends to initial data with zero Lelong numbers, paralleling the corresponding result on compact Kähler manifolds. The source presents it as a natural further direction and does not provide a resolution.