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Pokrovskiy conjectured that there is a function $f: \mathbb{N} \rightarrow\mathbb{N}$ such that any $2k$-strongly-connected tournament with minimum outand in-degree at least $f(k)$ is $k$-linked. In this paper, we show that any$(2k+1)$-strongly-connected tournament with minimum out-degree at least somepolynomial in $k$ is $k$-linked, thus resolving the conjecture up to theadditive factor of $1$ in the connectivity bound, but without the extraassumption that the minimum in-degree is large. Moreover, we show the conditionon high minimum out-degree is necessary by constructing arbitrarily largetournaments that are $(2.5k-1)$-strongly-connected but are not $k$-linked.

$(2k+1)$-connected tournaments with large minimum out-degree are $k$-linked