The linear connectivity conjecture for linked tournaments

A tournament is an orientation of a complete graph. A tournament is strongly $r$-connected when deleting fewer than $r$ vertices leaves a strongly connected digraph. A tournament is $k$-linked if, for every choice of $k$ distinct initial vertices and $k$ distinct terminal vertices, there are $k$ vertex-disjoint directed paths linking the prescribed pairs.

Linear connectivity conjecture for linked tournaments. There exists a constant $C>0$ such that, for every $k\in\mathbb{N}$, every strongly $Ck$-connected tournament is $k$-linked.

The paper proves a near-linear bound, namely strong $10^4k\log k$-connectivity implies $k$-linkedness, and notes that removing the logarithmic factor remains conjectural.