The zero residual Monge–Ampère mass conjecture

Let $D=B_1$ be the unit ball in ${\mathbb C}^n$, and let $u$ be a plurisubharmonic function on $B_1$ that is locally bounded outside the origin. The Monge–Ampère measure $(dd^c u)^n$ is assumed to be defined, and its atomic mass at the origin is the residual Monge–Ampère mass, normalized by

$$\tau_u(0)=\frac{1}{\pi^n}(dd^c u)^n(\{0\}).$$

Denote the Lelong number of $u$ at the origin by $\nu_u(0)$. Zero residual mass conjecture. If

$$\nu_u(0)=0,$$

then

$$\tau_u(0)=0.$$

Equivalently, a Dirac mass of $(dd^c u)^n$ at the origin should imply that $u$ has a positive Lelong number there. This is the zero mass conjecture raised by Guedj and Rashkovskii. The paper proves it for $S^1$-invariant plurisubharmonic functions in ${\mathbb C}^2$, while the general question remains open.