The conjecture on the maximal order of vanishing of torus Laplacian eigenfunctions

Let $d\geq 2$. Let $\Gamma_d(\lambda)$ denote the maximal order of vanishing of a Laplacian eigenfunction on the $d$-dimensional torus with frequency parameter $\lambda$, and let $\mathcal{N}_d(\lambda)$ denote the corresponding eigenvalue multiplicity. Maximal vanishing-order conjecture.

$$\Gamma_d(\lambda) \sim_d \mathcal{N}_d(\lambda)^{\frac{1}{d-1}}.$$

The examples constructed in the paper provide the proposed highest possible order of vanishing, and the conjecture asserts that their order is optimal. The source indicates that strong evidence is available and refers to a heuristic discussion, but does not state a resolution.