The full support eigenfunctions conjecture for metric graphs

Let $(\Gamma,\boldsymbol{\ell})$ be a metric graph, and let an eigenfunction $f$ have full support when it does not vanish identically on any edge:

$$f|_{e_j}\not\equiv 0\quad\text{for all }j=1,2,\ldots,N.$$

Full support eigenfunctions conjecture. For any metric graph $(\Gamma,\boldsymbol{\ell})$, and any choice of a complete orthonormal sequence of eigenfunctions, there are infinitely many eigenfunctions with full support.

This conjecture concerns the prevalence of eigenfunctions that are nonzero on every edge. The source attributes it to Conjecture 4.3 of Hofmann, Kenda, Mugnolo and Plümer and gives no resolution.