Carousel extremal tournament conjecture for inducibility of $C_k$

Let $C_k$ be the $k$-vertex carousel tournament, and let $C_n$ be the carousel tournament on $n$ vertices. For a tournament $T$, write $I(C_k,T)$ for the number of induced copies of $C_k$ in $T$. Carousel extremal tournament conjecture. For all $k\ge 5$ and $n\ge k$, the unique $n$-vertex tournaments maximizing $I(C_k,T)$ are the tournaments $C_n$. This would extend the proved $C_4$ extremal result and the flag-algebra evidence for $C_5$, $C_6$, $C_7$, and $C_8$; the conjecture remains open in general.