Pokrovskiy's semidegree conjecture for linked tournaments

Let $k\in\mathbb{N}$. A tournament is a digraph obtained by orienting each edge of a complete graph in exactly one direction. For a tournament $D$, let $\delta^+(D)$ and $\delta^-(D)$ denote its minimum out-degree and minimum in-degree, respectively, and let

$$\delta^0(D)=\min\{\delta^+(D),\delta^-(D)\}$$

denote its minimum semidegree. A tournament is $k$-linked if every $2k$ distinct vertices can be paired in the prescribed order by $k$ pairwise vertex-disjoint directed paths.

Pokrovskiy's conjecture. For every $k\in\mathbb{N}$, there exists an integer $h(k)$ such that every $2k$-connected tournament with $\delta^0(D)\geq h(k)$ is $k$-linked.

The conjecture was disproved by a family of $2k$-connected tournaments with sufficiently large order and minimum semidegree that are not $k$-linked. Thus connectivity at least $2k+1$ is necessary when the minimum semidegree is prescribed as a function of $k$.