The inverse-linear bound for the exponent of random k-majority tournaments

Let $\mathcal{G}(n,k)$ be the probability space of $k$-majority tournaments on vertex set $[n]$ obtained by uniformly choosing, with replacement, $2k-1$ linear orders of $[n]$. Let $X(n,k)$ be the maximum $t$ such that a transitive tournament $T_t$ occurs in $G\sim\mathcal{G}(n,k)$, and define $r_k$ as the infimum over all $\alpha\in\mathbb{R}$ such that $\mathbb{E}[X(n,k)]\le n^\alpha$ for all sufficiently large $n$. Inverse-linear exponent conjecture.

$$r_k=O(1/k).$$

The bound would give a substantially stronger estimate for the expected size of the largest transitive subtournament in random $k$-majority tournaments and, as noted in the source, would improve the upper bound for $f_k(n)$. The paper establishes $r_k<1$ for every $k$ and gives $r_2\le 2/3$, but the stated inverse-linear bound is not resolved here.