Deterministic sign-balance conjecture for Laplace eigenfunctions

Let $\mathcal{M}$ be a chaotic smooth compact $d$-manifold, and let $\{(\varphi_{j},\lambda_{j})\}_{j\ge 1}$ be the corresponding sequence of Laplace eigenfunctions and eigenvalues. Deterministic sign-balance conjecture. Along a density-$1$ subsequence, the $\varphi_{j}$ are sign-balanced above the scale

$$\frac{(\log{T})^{\frac{1}{d-1}+o(1)}}{T}.$$

This conjecture proposes the deterministic analogue of the paper's random sign-balance results in a generic chaotic setting. Its validity for chaotic manifolds and the stated logarithmic scale remain open.