Yuster's conjecture on inversion number for 3-decycling sets

Let $T$ be an $n$-vertex tournament. An inversion reverses all edges whose endpoints lie in a specified vertex set, and ${\rm inv}_k(T)$ is the minimum length of a sequence of inversions using sets of size at most $k$ that transforms $T$ into the transitive tournament. Define

$${\rm inv}_k(n)=\max_T {\rm inv}_k(T),$$

where the maximum is over all $n$-vertex tournaments. A tournament is quasi-random if it satisfies the standard equivalent properties of a random tournament asymptotically. Yuster's conjecture.

$${\rm inv}_3(n)=(1+o(1))\frac{n^2}{12}.$$

Moreover, an $n$-vertex tournament $T$ satisfies

$${\rm inv}_3(T)=(1+o(1))\frac{n^2}{12}$$

if and only if it is quasi-random. The lower bound follows from the feedback-edge-set interpretation of ${\rm inv}_2(T)$, and random tournaments attain the corresponding bound with high probability. The conjecture asserts both asymptotic tightness for $k=3$ and uniqueness of quasi-random extremal tournaments.