Hippchen's conjecture on intersections of longest paths

Let $G$ be a $k$-connected graph. A longest path in $G$ is a path with the maximum possible number of vertices among all paths in $G$.

Hippchen's conjecture. Any two longest paths in $G$ share at least $k$ vertices.

The conjecture concerns how the vertex-connectivity of a graph controls intersections among its longest paths. The source reports that Gutiérrez proved it for $k\leq 4$ and for $k\geq \frac{n-2}{3}$, while the paper improves these ranges to $k=5$ and $k\geq \frac{n+2}{5}$; the parser provides no evidence that the full conjecture has been resolved.