Lichiardopol's conjecture on disjoint cycles in tournaments

In 2010, N. Lichiardopol conjectured for $q\ge 3$ and $k\ge 1$ that any tournament with minimum out-degree at least $(q-1)k-1$ contains $k$ disjoint cycles of length $q$. We prove this conjecture for $q\ge 5$. Since it is already known to hold for $q\le 4$, this completes the proof of the conjecture.