Universal error-function edge kernel conjecture for pluricomplex polynomial Bergman kernels

Let $d\geq 1$, let $\mathscr Q:\mathbb C^d\to\mathbb R$ be $C^2$ and strictly plurisubharmonic, and let $S_{\mathscr Q}$ be its droplet with smooth boundary. For $z_0\in\partial S_{\mathscr Q}$, denote by $\vec n(z_0)\in\mathbb C^d$ the outward unit normal vector at $z_0$. Universal edge-kernel conjecture. The normalized kernel has the local limit

$$\lim_{n\to\infty} \frac1{\det n \partial\bar\partial\mathscr Q(z_0)} \mathscr K_n\left(z_0+\frac{\vec n(z_0)\xi}{\sqrt{n \partial\bar\partial\mathscr Q(z_0)}},z_0+\frac{\vec n(z_0) \eta}{\sqrt{n \partial\bar\partial\mathscr Q(z_0)}}\right)$$ $$\equiv \frac12 \exp\left(\xi\overline\eta-\frac{|\xi|^2+|\eta|^2}2\right) \operatorname{erfc}\left(\frac{\xi+\overline\eta}{\sqrt 2}\right),$$

locally uniformly for $\xi,\eta\in\mathbb C$. This conjecture is the higher-dimensional analogue of the known one-dimensional universal local edge scaling limit; the paper presents evidence from the pluripotential elliptic Ginibre model and related results, but the asserted universality in the stated generality remains open.