Gir2o–Snyder–Popielarz linear out-degree conjecture for linked tournaments

Let $k$ be a positive integer. A tournament is a complete graph with exactly one orientation assigned to each edge. A digraph is $k$-linked if every $2k$-tuple of distinct vertices $x_1,\ldots,x_k,y_1,\ldots,y_k$ can be joined by $k$ pairwise vertex-disjoint directed paths, with the $i$th path starting at $x_i$ and ending at $y_i$. Let $\delta^+(D)$ denote the minimum out-degree of a tournament $D$.

Gir2o–Snyder–Popielarz conjecture. There exists a constant $C>0$ such that every $(2k+1)$-connected tournament with minimum out-degree at least $Ck$ is $k$-linked.

The source reports this as an unresolved conjecture. Its significance is that the paper proves the same connectivity condition with a polynomial minimum out-degree bound, while the conjecture asks for a linear bound in $k$.