Many short leaf-leaf path lengths in 1-3 trees

A 1-3 tree is a tree whose vertices have degrees $1$ or $3$, and a leaf-leaf path is a path with leaf endpoints. Many-short-paths conjecture. There is a constant $\alpha>0$ and a function $C(n)$ tending to infinity such that every $1$-$3$ tree of order $n$ contains at least

$$\alpha C(n)$$

distinct leaf-leaf path lengths between $0$ and $C(n)$. This refines the preceding path-count question by requiring a positive proportion of all lengths up to $C(n)$. The paper's constructions rule out guaranteeing some sufficiently large individual path lengths, but leave open whether many short lengths must nevertheless occur.