Linial–Morgenstern conjecture on minimizing 4-cycles in tournaments

Let $T$ be a tournament on $n$ vertices, and let $c_3$ and $c_4$ denote its normalized numbers of cyclic triples and cyclic $4$-cycles, respectively. For a transitive tournament $T_m$ and weights $w_1,\ldots,w_m$, form the random blow-up by replacing vertex $i$ with a class of asymptotic relative size $w_i$, orienting edges between classes according to $T_m$, and choosing each class internally at random. Linial–Morgenstern conjecture. For fixed $c_3$, the minimum of $c_4$ is attained by a random blow-up of a transitive tournament $T_m$. Among such blow-ups with the prescribed $c_3$, the minimum is attained by taking $m$ as small as possible and $w_1=w_2=\cdots=w_{m-1}\geq w_m$. This conjecture proposes an extremal description of tournaments minimizing cyclic $4$-cycles subject to a prescribed density of cyclic triples; the cited passage does not state whether it has been resolved.