The slow-decay conjecture for bipartite MaxLA on free trees

Let $n$ be the number of vertices and let $\operatorname{propbip}(n)$ denote the proportion of free $n$-vertex trees for which a maximal bipartite arrangement is a maximum arrangement. For functions of $n$, $f(n)=\Theta(g(n))$ means that each is bounded above and below by a positive constant multiple of the other for all sufficiently large $n$.

Slow-decay conjecture. The proportion decays according to

$$\operatorname{propbip}(n)=\Theta(n^{-b})$$

for some $b\in(0,1)$.

This conjecture concerns the fallback scenario in which the positive limiting-proportion conjectures fail. A power-law decay with exponent below one would mean that the proportion remains relatively large for finite orders, supporting bipartite MaxLA as an approximation or practical method; the paper provides statistical evidence but no proof.