The positive bipartite MaxLA proportion conjecture for free trees

Let $n$ be the number of vertices, let $\mathcal{T}_n$ be the set of all free $n$-vertex trees, and let $\mathcal{M}_n$ be the set of those trees for which a maximal bipartite arrangement is also a maximum arrangement. Define

$$\operatorname{propbip}(n)=\frac{|\mathcal{M}_n|}{|\mathcal{T}_n|}.$$

Weak bipartition conjecture. The same limiting assertion as in the strong bipartition conjecture holds with merely $c>0$; equivalently, the asymptotic proportion is positive.

This is a weaker form of the preceding conjecture, intended to remain plausible even if the observed limiting value near $0.5$ is not sustained. It would still imply that bipartite MaxLA solves a positive proportion of free trees for large $n$; the paper gives empirical support but no proof.