The quadratic linkedness conjecture for edge-disjoint Hamilton cycles

A tournament is an orientation of a complete graph, a Hamilton cycle is a consistently oriented cycle containing all vertices, and a tournament is $k$-linked if it contains vertex-disjoint directed paths joining every prescribed set of $k$ initial vertices to every prescribed set of $k$ terminal vertices in the prescribed pairing.

Quadratic linkedness conjecture. There exists $C'>0$ such that, for every $k\in\mathbb{N}$, every $C'k^2$-linked tournament contains $k$ edge-disjoint Hamilton cycles.

The paper proves the corresponding bound with an additional logarithmic factor, so removal of that factor remains open in the supplied text.