Half-line hyperuniformity conjecture for Schrödinger eigenvalue counts

Let $H$ be the Schrödinger operator considered in the paper, let $N(\lambda)$ be its eigenvalue counting function, and let $R_{\mathrm h}(\lambda)$ denote the half-line hyperuniformity ratio. For dimensions $d=1,2,3$, the half-line hyperuniformity conjecture asserts that

$$R_{\mathrm h}(\lambda)\to0$$

as $\lambda\to\infty$ at an explicit rate depending on $d$. This is motivated by the paper's hyperuniformity results and an informal Abelian/Tauberian relation; the required asymptotics appear to depend on obtaining variance estimates for $N(\lambda)$, and the conjecture remains open.