The total-order conjecture for trees under the independence-polynomial order

Let $\succeq$ be the order on graphs defined by comparing their independence polynomials, and let $\mathcal{T}_n$ be the set of trees with $n$ vertices. Total-order conjecture. For any two trees $T_1,T_2\in\mathcal{T}_n$,

$$T_1\succeq T_2\quad\text{or}\quad T_2\succeq T_1.$$

The claim would make all trees of a fixed order comparable under this graph order. The paper notes that the order is not total on all graphs and gives partial ordering results for trees, but does not prove totality for all trees.