Eventual equality of tournament counting exponents

For a positive integer $k$, let $c(k)$ be the exponent for counting copies of an arbitrary $k$-vertex tournament, and let $c^*(k)$ be the exponent for counting copies of the transitive tournament $K_k$. Eventual-equality conjecture. For all sufficiently large $k$ it holds that $c(k)=c^*(k)$. This strengthens the preceding inequality conjecture; the paper presents it as an open direction, with no resolution supplied.