Continuity conjecture for the non-deterministic density of states measure

Let $u$ and $u^{ND}$ be the limiting density of states measures, and write $u- u^{ND}$ for their difference, denoted $\mu-\mu^{ND}$ in the source. A measure is continuous when it does not charge any point.

Continuity conjecture. The measure $\mu-\mu^{ND}$ is continuous, i.e. it does not charge any point.

This conjecture concerns the regularity of the density of states after removing the non-deterministic component; the source relates it to results on Lelong numbers of the Green current. Its resolution is not given in the supplied text.