Sarnak–Chiu conjecture on the height of flat tori

Let $L=C\mathbb{Z}^n\subset\mathbb{R}^n$ be a lattice with Gram matrix $G=C^{\mathsf T}C$ and associated flat torus $T_L=\mathbb{R}^n/L$. Normalize to unit volume, so $\det G=1$. Its squared first minimum is

$$\lambda_1(L)^2=\min_{m\in\mathbb{Z}^n\setminus\{0\}}m^{\mathsf T}Gm.$$

Let $\Delta_G$ be the nonnegative Laplacian on $T_L$, with spectral zeta function

$$\zeta_{\Delta_G}(s)=\sum_{\lambda\in\operatorname{Spec}(\Delta_G)\setminus\{0\}}\lambda^{-s},$$

and define the height by $h(G)=\zeta'_{\Delta_G}(0)$. Sarnak–Chiu conjecture. Among all $n$-dimensional flat tori of volume one, the height is minimized by a torus corresponding to a lattice maximizing the length of its shortest nonzero vector. At fixed covolume, maximizing this shortest-vector length is equivalent to maximizing the density of the associated lattice sphere packing; the conjecture therefore relates an analytic optimization problem for the regularized determinant to the lattice sphere-packing problem. Its resolution is not supplied here, so its status remains open.