Mader's nonseparating dipath conjecture

Let $k$ and $m$ be positive integers, and let $D$ be a $k$-connected digraph. Denote its minimum outdegree and indegree by $\delta^+(D)$ and $\delta^-(D)$, and define its minimum semi-degree by

$$\delta(D)=\min\{\delta^+(D),\delta^-(D)\}.$$

A dipath $P$ has order $m$ when it contains $m$ vertices, and $\kappa(D)$ denotes vertex-connectivity.

Mader's nonseparating dipath conjecture. If

$$\delta(D)\geq 2k+m-1,$$

then $D$ has a dipath $P$ of order $m$ such that

$$\kappa(D-V(P))\geq k.$$

This is the digraph analogue of the nonseparating path conjecture. At the time of the paper, it had been verified only for $m=1$, and for $m=2$ with $k=1$; the general assertion remains open.