Central limit conjecture for the Neumann surplus

Let $\{\Gamma^{(m)}\}_{m=1}^{\infty}$ be standard quantum graphs with rationally independent edge lengths. Let $\beta^{(m)}$ and $\left|\partial\Gamma^{(m)}\right|$ be the first Betti number and boundary size of $\Gamma^{(m)}$, and let $\omega^{(m)}$ be its Neumann surplus random variable. Assume

$$\lim_{m\rightarrow\infty}\left(\beta^{(m)}+\left|\partial\Gamma^{(m)}\right|\right)=\infty.$$

Neumann surplus central limit conjecture. Then

$$\frac{\omega^{(m)}-\mathbb{E}\left[\omega^{(m)}\right]}{\sqrt{\mathbb{V}\left[\omega^{(m)}\right]}}\xrightarrow[m\rightarrow\infty]{\mathcal{D}}N(0,1),$$

where convergence is in distribution and $N(0,1)$ is the standard normal distribution. This predicts universal Gaussian fluctuations for Neumann surplus along every such increasing graph sequence, but the supplied text gives no resolution.