Infinite minimal Laplacian-tree conjecture

Let $\mu$ be a Laplacian eigenvalue of some tree $T$, with $\mu\neq 1$. Let $\mathcal{LT}_{\mathrm{min},\mu}^{0}$ and $\mathcal{LT}_{\mathrm{min},\mu}^{1}$ denote the two classes of minimal Laplacian trees used in the source, and let $\operatorname{Norm}(\mu)$ denote the stated norm. Infinite minimal Laplacian-tree conjecture. If $\operatorname{Norm}(\mu)=1$, then

$$|\mathcal{LT}_{\mathrm{min},\mu}^{0}|=|\mathcal{LT}_{\mathrm{min},\mu}^{1}|=+\infty.$$

If $\operatorname{Norm}(\mu)>1$, then

$$|\mathcal{LT}_{\mathrm{min},\mu}^{0}|=+\infty.$$

The claim predicts infinitely many minimal trees in the indicated classes, while the preceding discussion states that the class with superscript $1$ is empty when $\operatorname{Norm}(\mu)>1$. The supplied text gives no resolution status.