Ore-type oriented Hamilton cycle discrepancy conjecture

Let $D$ be an oriented graph on $n\ge 3$ vertices with minimum degree $\delta$. For vertices $u,v$, write $d(u)$ and $d(v)$ for their degrees, and call them non-adjacent if neither directed arc joins them. For an oriented Hamilton cycle $C$, let $\sigma_{\max}(C)$ be the larger of its numbers of forward and backward arcs. Ore-type discrepancy conjecture. If

$$d(u)+d(v)\ge n$$

for each pair of non-adjacent vertices $u$ and $v$, then there exists a Hamilton oriented cycle $C$ in $D$ such that

$$\sigma_{\max}(C)\ge \max\{\delta,n-\delta\}.$$

This conjecture extends the cited Dirac-type result from a minimum-degree condition to an Ore-type condition. It is presented as open; the authors state that they could not prove it but provide supporting results, and the bound is sharp in the discussed cases.