Asymptotic $L^p$ symmetry conjecture for positive and negative parts of eigenfunctions

Let $(M,g)$ be a given Riemannian manifold, let $\psi_{\lambda}$ be a real Laplace eigenfunction, and let $1<p<\infty$. For a set $A\subseteq M$, write $\chi_A$ for its characteristic function, and let $\|\cdot\|_p$ denote the $L^p$ norm. Asymptotic $L^p$ symmetry conjecture.

$$\frac{\|\psi_{\lambda}\chi_{\{\psi_{\lambda}>0\}}\|_{p}}{\|\psi_{\lambda}\chi_{\{\psi_{\lambda}<0\}}\|_{p}}\longrightarrow 1$$

as $\lambda\rightarrow\infty$ for $1<p<\infty$ on the given manifold. This question concerns whether the positive and negative parts become asymptotically equal in $L^p$ norm. The cited source presents it as unclear, and the paper does not report a resolution.