Central limit conjecture for zeros of random Newman and Littlewood polynomials

Let $\mathcal N_n$ and $\mathcal L_n$ denote the relevant families of Newman and Littlewood polynomials of degree $n$, and let $N$ and $\widetilde N$ count zeros in the open unit disk with and without the specified unimodular-zero restriction, respectively. Central limit conjecture. For $f\in\mathcal N_{2n+1}$ and $g\in\mathcal L_n$, each standardized variable

$$\frac{N(f)-\mathbb E N(f)}{\sqrt{\operatorname{var}(N(f))}},\quad \frac{\widetilde N(f)-\mathbb E\widetilde N(f)}{\sqrt{\operatorname{var}(\widetilde N(f))}},$$ $$\frac{N(g)-\mathbb E N(g)}{\sqrt{\operatorname{var}(N(g))}},\quad \frac{\widetilde N(g)-\mathbb E\widetilde N(g)}{\sqrt{\operatorname{var}(\widetilde N(g))}}$$

converges in distribution to $Z(0,1)$ as $n\to\infty$. The conjecture is motivated by the numerical distributions shown in the paper, and no proof or disproof is supplied in the given context.