Common minimal-tree conjecture for totally real algebraic integers

Let $k\in\mathbb{N}$ and let $\lambda_1,\ldots,\lambda_k\in\mathtt{TRAI}\setminus\{0\}$. For each $\lambda_i$, let $\mathcal{T}_{\mathrm{min},\lambda_i}$ denote the set of trees having a nowhere-zero $\lambda_i$-eigenvector. Common minimal-tree conjecture.

$$\mathcal{T}_{\mathrm{min},\lambda_1}\nsubseteq\mathcal{T}_{\mathrm{min},\lambda_2},\qquad \bigcap_{i=1}^{k}\mathcal{T}_{\mathrm{min},\lambda_i}\neq\emptyset.$$

The conjecture asserts both that the first two minimal-tree families are not nested and that one tree can be minimal for all the specified nonzero eigenvalues. The supplied text gives no resolution status.