The recursive upper-bound conjecture for tournament counting

For each positive integer $k$, let $c(k)$ denote the exponent for counting copies of an arbitrary $k$-vertex tournament. Recursive counting conjecture. For all $k \ge 3$ it holds that $c(k) \le c(k-1)+1$. The paper notes that this inequality is not obvious, unlike the corresponding inequalities for $c^*(k)$ and for a fixed tournament; no resolution is supplied.