Nash-Williams' directed degree-sequence conjecture

Let $D$ be a strongly connected digraph on $n$ 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 degree-sequence conjecture. If, for every integer $i<n/2$, either $d_i^+\geq i+1$ or $d_{n-i}^-\geq n-i$, and either $d_i^-\geq i+1$ or $d_{n-i}^+\geq n-i$, then $D$ contains a Hamilton cycle. This is the proposed directed analogue of Chvátal's theorem; the source describes the asymptotic version as a consequence of its results, while the exact conjecture is not resolved here.