The conjecture is proved individually for up to 13 runners, most recently n = 10 through 13 in 2025 and 2026, and Tao proved an improved general asymptotic bound in 2018, but no argument covers every n at once.
What would resolve this A proof, or a counterexample, covering every number of runners n rather than a further specific value.
OpenCombinatoricsLonely Runner Conjecture (Wikipedia)