  Pokrovskiy conjectured that there is a function $f: \mathbb{N} \rightarrow
\mathbb{N}$ such that any $2k$-strongly-connected tournament with minimum out
and 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 some
polynomial in $k$ is $k$-linked, thus resolving the conjecture up to the
additive factor of $1$ in the connectivity bound, but without the extra
assumption that the minimum in-degree is large. Moreover, we show the condition
on high minimum out-degree is necessary by constructing arbitrarily large
tournaments that are $(2.5k-1)$-strongly-connected but are not $k$-linked.
