The finite-frequency zero mass conjecture for plurisubharmonic functions

Let $u\in {\mathcal P}(B_1)$ be a plurisubharmonic function whose alternating part has finite frequencies, meaning that near the origin, in a proper complex Hopf-coordinate, it can be written as

$$u = u_s+\sum_{k=1}^m\left(\cos k\theta\cdot v_k+\sin k\theta\cdot w_k\right).$$

Here $u_s$ is the $S^1$-invariant part, $\theta$ is the argument in the $S^1$-direction, and $v_k,w_k$ are circular symmetric functions. Let $\nu_u(0)$ denote the Lelong number and $\tau_u(0)$ the residual Monge–Ampère mass at the origin. Finite-frequency zero mass conjecture. If $u$ has finite frequencies in its alternating part, then an upper-bound estimate for its Monge–Ampère mass should hold, and

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

This is proposed as a weaker version of the zero mass conjecture, motivated by the established uniformly directional Lipschitz result for functions that separate the $S^1$-direction. The conjectured upper-bound estimate for general finite-frequency functions remains unresolved in the supplied text.