Smilansky's nodal universality conjecture for quantum graphs

Let $\left\{ \Gamma^{\left(\beta\right)}\right\} _{\beta\nearrow\infty}$ be any sequence of standard graphs, with first Betti numbers $\beta\to\infty$. Choosing arbitrary rationally independent edge lengths for each $\Gamma^{\left(\beta\right)}$, let $\sigma^{\left(\beta\right)}$ denote its nodal surplus random variable.

Nodal universality conjecture. The variance has linear growth

$$\operatorname{var}\left(\sigma^{(\beta)}\right) \sim \beta$$

and, as $\beta\to\infty$,

$$\frac{\sigma^{\left(\beta\right)}-\frac{\beta}{2}} {\sqrt{\operatorname{var}\left(\sigma^{(\beta)}\right)}} \xrightarrow{\ \mathcal{D}\ } N(0,1),$$

where convergence is in distribution and $N(0,1)$ is the standard normal distribution. This conjecture concerns the universality of nodal-surplus statistics as the complexity of a quantum graph grows. It has been rigorously established only for certain families of quantum graphs, so the stated generality remains open.