Universal multivariate error-function edge kernel conjecture

Let $d\geq 1$, let $\mathscr Q:\mathbb C^d\to\mathbb R$ be $C^2$ and strictly plurisubharmonic, and suppose that its droplet $S_{\mathscr Q}$ has smooth boundary. For each $z_0\in\partial S_{\mathscr Q}$, let $\mathscr U(z_0)$ be a unitary matrix. Universal multivariate edge-kernel conjecture. There is a choice of $\mathscr U(z_0)$ such that

$$\lim_{n\to\infty} \frac1{\det n \partial\bar\partial\mathscr Q(z_0)} \mathscr K_n\left(z_0+\frac{\mathscr U(z_0)\xi}{\sqrt{n \det\partial\bar\partial\mathscr Q(z_0)}},z_0+\frac{\mathscr U(z_0) \eta}{\sqrt{n \det\partial\bar\partial\mathscr Q(z_0)}}\right)$$ $$\equiv \frac12 \exp\left(\xi\cdot\eta-\frac{|\xi|^2+|\eta|^2}2\right) \operatorname{erfc}\left(\sum_{k=1}^d \frac{\xi_k+\overline{\eta_k}}{\sqrt {2d}}\right),$$

locally uniformly for $\xi,\eta\in\mathbb C^d$. This is the paper's proposed second universal edge behavior, obtained after a unitary rotation to remove the geometric dependence on the boundary normal; the paper gives strong evidence in specific models and general rotational results, while the conjecture remains open.