The symmetry conjecture for spherical harmonics

Let $\{\psi_{\lambda}\}$ be a sequence of spherical harmonics on the sphere, with eigenvalue parameter $\lambda$. Consider the ratio of the volumes of their positive and negative sets.

Symmetry conjecture. The limit

$$\frac{\operatorname{vol}(\{x\in M:\psi_{\lambda}(x)>0\})}{\operatorname{vol}(\{x\in M:\psi_{\lambda}(x)<0\})}\longrightarrow 1$$

holds as $\lambda$ grows to infinity.

This is presented as a partial result toward the broader symmetry conjecture, which is refuted in general but remains valid in some special settings, including the two-dimensional flat torus. The source places this statement in the context of spherical harmonics and an application of the paper's theorem for $f(z)=\cos(z)$.