Conjecture on the dimension of pluriharmonic measure in complex domains

Let $\Omega$ be a domain in $\mathbb{C}^n$, and let pluriharmonic measure on $\Omega$ have dimension defined as the minimal Hausdorff dimension of a set of full measure.

Pluriharmonic-measure dimension conjecture. The dimension of pluriharmonic measure of domains in $\mathbb{C}^n$ is at most

$$2n-1.$$

The theorem proved in the paper establishes this bound for pluriharmonic measure arising from regular polynomial endomorphisms, and the conjecture extends the bound to domains in general. The estimate is sharp, as shown by area measure on the unit sphere in $\mathbb{C}^n$; the source gives no resolution of the general conjecture.