Nash-Williams' Chvátal-type conjecture for Hamilton cycles in digraphs

Let $G$ be a strongly connected digraph on $n\geq 3$ vertices, and let $d^+_1\leq\cdots\leq d^+_n$ and $d^-_1\leq\cdots\leq d^-_n$ be its ordered outdegree and indegree sequences. Nash-Williams' conjecture. If, for every $i<n/2$,

$$d^+_i\geq i+1\text{ or }d^-_{n-i}\geq n-i,$$

and

$$d^-_i\geq i+1\text{ or }d^+_{n-i}\geq n-i,$$

then $G$ contains a Hamilton cycle. This is a directed analogue of Chvátal's best-possible degree-sequence condition; its status is not resolved in the supplied source.