Oboudi's comparability conjecture for trees under independence-polynomial order

Let $T_1$ and $T_2$ be trees of order $n$. Write $T_1\triangleleft T_2$ when the independence polynomial order satisfies the strict relation, and $T_2\trianglelefteq T_1$ for the corresponding non-strict relation. Oboudi's comparability conjecture. Is it true that

$$T_1\triangleleft T_2\quad\text{or}\quad T_2\trianglelefteq T_1?$$

The question asks whether every pair of trees of the same order is comparable in this order; the paper presents an infinite family showing that the proposed comparability fails.