Gire3o–Popielarz–Snyder linear semidegree conjecture for linked tournaments

Let $k\in\mathbb{N}$ and let $T$ be a tournament. Write $\delta^+(T)$ for the minimum out-degree of $T$. A tournament is $k$-linked if every prescribed pairing of $2k$ distinct vertices can be joined by $k$ pairwise vertex-disjoint directed paths.

Gir~{a}o–Popielarz–Snyder conjecture. There exists a constant $C>0$ such that every $(2k+1)$-connected tournament with $\delta^+(T)\geq Ck$ is $k$-linked.

Gir~{a}o, Popielarz, and Snyder proved a polynomial sufficient bound, namely $\delta^+(T)\geq Ck^{31}$, while the conjectured linear dependence on $k$ remains open.