The full-support eigenfunction conjecture for compact metric graphs

Let $\mathcal{G}$ be a compact, connected metric graph. The Laplacian on $\mathcal{G}$ has natural vertex conditions, and an eigenfunction is said to have full support if it does not vanish identically on any edge. For an eigenfunction $\psi_n$, let $|\{\psi_n\neq 0\}|$ denote the total length of the edges on which it is not identically zero, and let $|\mathcal{G}|$ be the total length of the graph.

Full-support eigenfunction conjecture. There exists a choice of eigenfunctions $\psi_n$ forming an orthonormal basis of $L^2(\mathcal{G})$ and a subsequence $(n_k)\subset\mathbb{N}$ such that no $\psi_{n_k}$ vanishes identically on any edge of $\mathcal{G}$. Equivalently,

$$1\in\operatorname{acc}\left\{\frac{|\{\psi_n\neq 0\}|}{|\mathcal{G}|}:n\in\mathbb{N}\right\}.$$

The conjecture would imply that, for every compact connected metric graph, one can choose natural-Laplacian eigenfunctions with nodal-domain ratios having limit superior equal to $1$.