Frank's vertex-connected orientation conjecture

Let $G=(V,E)$ be a graph and let $k$ be a positive integer. A $k$-vertex-connected orientation is an orientation of $G$ that is $k$-vertex-connected. Frank's conjecture. $G$ has a $k$-vertex-connected orientation if and only if $|V|\geq k+1$ and $G-X$ is $2(k-|X|)$-edge-connected for all $X\subseteq V$ with $|X|\leq k-1$.

The conjecture was proved for $k=2$ by Thomassen, but was disproved for every $k\geq 3$ by Durand de Gevigney; deciding whether a graph has a $k$-vertex-connected orientation is NP-hard for every $k\geq 3$.