Conjecture on the maximal Nazarov–Sodin constant for symmetric measures

Let $\mathcal{P}_{Symm}$ be the family of symmetric probability measures on $\mathcal{S}^{1}$, and let $d_{\max}$ denote the maximal value attained by the Nazarov–Sodin constant $c_{NS}(\mu)$ for $\mu\in\mathcal{P}_{Symm}$. Let $\mu_{\mathcal{S}^{1}}$ be the uniform measure on $\mathcal{S}^{1}\subseteq\mathbb{R}^{2}$. Maximal Nazarov–Sodin constant conjecture. For $\mu\in\mathcal{P}_{Symm}$, the maximal value $d_{\max}$ is uniquely attained by $c_{NS}(\mu_{\mathcal{S}^{1}})$. In particular,

$$d_{\max}=c_{\text{RWM}}.$$

The conjecture identifies the uniform spectral measure, corresponding to the random plane wave, as the unique maximizer among symmetric measures. The source presents this as an expected statement; no resolution is given.