Improvements on Hippchen's conjecture

Let G be a k-connected graph on n vertices. Hippchen's Conjecture (2008) states that two longest paths in G share at least k vertices. Gutiérrez (2020) 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 by Gutiérrez in the affirmative.