Faudree–Schelp conjecture on path lengths in hamiltonian-connected graphs

Let $G$) be a hamiltonian-connected graph on $n$ vertices. For every pair of distinct vertices $u,v$ in $G$, a path between $u$ and $v$ has length $k$ for each integer $k$ satisfying

$$\frac{n}{2}\leq k\leq n-1.$$

Faudree–Schelp conjecture. Every such pair $u,v$ has a path of every length $k$ in this range. The conjecture was disproved by Thomassen, who constructed hamiltonian-connected graphs with pairs of vertices having no path of length $n-2$; the paper further gives cubic planar counterexamples.