Narins–Pokrovskiy–Szabó conjecture on leaf-to-leaf path lengths in 1–3 trees

A 1–3 tree is a tree in which every vertex has degree either $1$ or $3$. Narins–Pokrovskiy–Szabó conjecture. Every 1–3 tree $T$ of order $n$ has leaf-to-leaf paths of at least

$$\log(n+2)-1$$

distinct lengths. The source proves this claim as a consequence of a stronger theorem for arbitrary trees with bounded maximum degree, so the conjecture is solved.