Gnutzmann–Smilansky–Weber universal nodal-surplus Gaussian conjecture

Let $\{\Gamma_{\vec{l}}^{(\beta)}\}_{\beta\nearrow\infty}$ be any sequence of standard graphs parameterized by their first Betti numbers, with each graph having rationally independent edge lengths. Let $\sigma^{(\beta)}$ be the corresponding nodal surplus random variable. Gnutzmann–Smilansky–Weber's conjecture.

$$\frac{\sigma^{(\beta)}-\frac{\beta}{2}}{\sqrt{\operatorname{Var}(\sigma^{(\beta)})}}\xrightarrow[\beta\to\infty]{\mathcal{D}}N(0,1),$$

where the convergence is in distribution and

$$\operatorname{Var}(\sigma^{(\beta)})=\mathcal{O}(\beta).$$

This is the universal formulation of the nodal-statistics conjecture in terms of nodal surplus. The source proves it for trees of cycles, while the assertion for arbitrary such graph sequences remains open.