Improvements on Hippchen's Conjecture

Let $G$ be a $k$-connected graph on $n$ vertices. Hippchen’s Conjecture states that two longest paths in $G$ share at least $k$ vertices. Guti'errez recently proved the conjecture when $k\le 4$ or $k\ge \frac{n-2}{3}$. We improve upon both results; namely, we show that two longest paths in $G$ share at least $k$ vertices when $k=5$ or $k\ge \frac{n+2}{5}$. This completely resolves two conjectures of Guti'errez in the affirmative.