Nash-Williams' directed Pósa-type conjecture

Let $D$ be a digraph on $n\geq 3$ vertices, with nondecreasing out-degree sequence $d_1^+\leq\cdots\leq d_n^+$ and in-degree sequence $d_1^-\leq\cdots\leq d_n^-$. Nash-Williams' directed Pósa-type conjecture. If $d_i^+,d_i^-\geq i+1$ for every $1\leq i<(n-1)/2$, and, when $n$ is odd, additionally $d^+_{\lceil n/2\rceil},d^-_{\lceil n/2\rceil}\geq\lceil n/2\rceil$, then $D$ contains a Hamilton cycle. This is presented as a weakening of Nash-Williams' directed degree-sequence conjecture and would yield a directed analogue of Pósa's theorem; its resolution is not given in the source.