The 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

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

Strong bipartition conjecture. The limit exists and is a positive constant:

$$\lim_{n\to\infty}\operatorname{propbip}(n)=c,$$

with statistical analyses suggesting $c\approx 0.5$.

The conjecture predicts that a substantial proportion of free trees remain solvable by bipartite MaxLA asymptotically, which would give a linear-time method for a large class of trees. The paper presents exhaustive computations for small orders and sampling-based evidence for larger orders, but no proof of the limiting behavior is given.