Conjecture on the maximal Nazarov–Sodin constant for planar spectral measures

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

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

This is the corresponding maximality conjecture for arbitrary spectral measures supported in the unit ball. It is stated in relation to the symmetric-measure conjecture above, and the source gives no resolution.