The odd-bridge conjecture for 2-linear trees

Let $T$ be a $2$-linear tree, and let its bridge be the distinguished path joining its two linear components. Write $\operatorname{Max}(T)$ for the set of maximum arrangements of $T$, and $\operatorname{MaxBip}(T)$ for the set of maximal bipartite arrangements of $T$.

Odd-bridge conjecture. If the bridge of $T$ has odd length, then no maximal non-bipartite arrangement is a maximum arrangement; equivalently,

$$\operatorname{Max}(T)=\operatorname{MaxBip}(T).$$

The conjecture is motivated by exhaustive data showing that odd-bridge $2$-linear trees of the tested orders are maximizable only by bipartite arrangements. A proof for all such trees is not supplied.