Narins–Pokrovskiy–Szabó conjecture on short leaf-to-leaf path lengths

A 1–3 tree is a tree in which every vertex has degree either $1$ or $3$. For a tree, a leaf-to-leaf path is a path whose endpoints are leaves. Narins–Pokrovskiy–Szabó conjecture. There exist a constant $\alpha>0$ and a function $N=N(n)$ with $N(n)\to\infty$ as $n\to\infty$ such that every 1–3 tree of order $n$ contains at least $\alpha N$ distinct leaf-to-leaf path lengths between $0$ and $N$. The source gives upper and lower bounds with different exponents for this short-length problem, so the conjecture remains open.