Symmetry conjecture for positive and negative volumes of Laplace eigenfunctions

Let $(M,g)$ be a smooth Riemannian manifold, and let $\psi_{\lambda}$ be a real Laplace eigenfunction satisfying

$$-\Delta_g\psi_{\lambda}=\lambda\psi_{\lambda}.$$

Write $\operatorname{vol}$ for Riemannian volume. 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 strengthens the known uniform quasi-symmetry result. It is disproved for $n$-dimensional tori with $n\geq 3$, while it holds on the two-dimensional torus; whether it holds for other two-dimensional manifolds remains open.