The conjectured sharp lower bound for observability constants on high-dimensional tori

Let $m_d(r)$ be the optimal observability constant for Laplacian eigenfunctions on the $d$-dimensional torus observed on a ball of radius $r$. For $d\geq 4$, let $C_d$ and $C'_d$ be the constants appearing in the conjectured matching lower and upper bounds. Sharp observability-bound conjecture. If $d\geq 4$, then

$$\exp\bigl\{-C_d r^{3-d}\bigr\} \lesssim m_d(r) \lesssim \exp\bigl\{-C'_d r^{3-d} \bigr\}.$$

This conjecture says that the optimal lower bound for $m_d(r)$ matches the upper bound obtained in the paper. The source reports strong evidence for the conjectures in this section, but gives no resolution of this claim.