Quadratic additive bound conjecture for sparse spanning strongly connected subgraphs of tournaments

Let $k$ be a positive integer, and let $T$ be a strongly $k$-connected tournament on $n$ vertices. A strongly $k$-connected spanning subgraph is a spanning subgraph with strong vertex-connectivity at least $k$. Quadratic additive bound conjecture. There is a constant $C>0$ such that $T$ contains a strongly $k$-connected spanning subgraph $D$ with at most

$$kn+Ck^2$$

arcs. The paper proves an upper bound of $kn+750k^2\log_2(k+1)$ arcs and states that reducing the additive term to $O(k^2)$ is conjectured; the supplied text gives no resolution of this conjecture.