Bipartitions of highly connected tournaments

We show that if $T$ is a strongly ${10}^9{k}^6\log (2k)$-connected tournament, there exists a partition $A,B$ of $V(T)$ such that each of $T[A]$, $T[B]$ and $T[A,B]$ is strongly $k$-connected. This provides tournament analogues of two partition conjectures of Thomassen regarding highly connected graphs.