The conjecture of $\mathbb{Q}$-independent spectrum for metric graphs

Let $\Gamma$ be a metric graph with $N$ edges and edge-length vector $\boldsymbol{\ell}\in\mathbb{R}_{+}^{N}$. Write the non-zero square-root eigenvalues in increasing order as

$$k_{1}\le k_{2}\le k_{3}\le\cdots\nearrow\infty.$$

$\mathbb{Q}$-independent spectrum conjecture. For every graph $\Gamma$, possibly except for some pathological cases, there is a generic set $G\subset\mathbb{R}_{+}^{N}$ such that, for every $\boldsymbol{\ell}\in G$, the spectrum $\operatorname{spec}(\Gamma,\boldsymbol{\ell})$ is linearly independent over $\mathbb{Q}$. Equivalently, for every $n\in\mathbb{N}$ and every $\boldsymbol{q}=(q_{1},\ldots,q_{n})\in\mathbb{Q}^{n}\setminus\{0\}$,

$$\sum_{j=1}^{n}k_{j}q_{j}\ne0.$$

Kurasov and Sarnak established that the spectrum has infinite dimension over $\mathbb{Q}$ when the edge lengths are $\mathbb{Q}$-independent; the conjecture asks for full linear independence for generic edge lengths. The source presents this as future work and gives no resolution.