Weyl-model asymptotic conjecture for zeros of random harmonic polynomials

Let $H_{n,m}(z)=p(z)+q(\overline{z})$ be a random harmonic polynomial in the Weyl model, with $p$ and $q$ analytic complex polynomials of degrees $n$ and $m$, respectively. Let $\mathcal{N}_{H}(\mathbb{C})$ denote the number of zeros of $H_{n,m}$ in the complex plane, and let $\mathbb{E}$ denote expectation over the Weyl ensemble. Weyl-model zero-count conjecture. The expected number of zeros satisfies

$$\mathbb{E}\mathcal{N}_{H}(\mathbb{C})\sim \frac{1}{3}m^{3/2}+n+O(\sqrt{n}).$$

The paper proves growth-order results when $m=\alpha n+O(1)$ and linear growth in $n$ when $m$ is fixed, but the displayed asymptotic is presented as a conjecture and its full validity remains unresolved.