Generalized oriented Hamilton cycle discrepancy conjecture via s-star

Let $D$ be an oriented graph on $n\ge 3$ vertices. For distinct vertices $u,v$, call them non-adjacent if neither directed arc joins them, and define

$$s^*(D)=\min\{d(u)+d(v)-n:u\ne v\in V(D),\ \{uv,vu\}\cap A(D)=\emptyset\}$$

when $D$ has a pair of non-adjacent vertices; set $s^*(D)=n-2$ when $D$ is a tournament. For an oriented Hamilton cycle $C$, let $\sigma_{\max}(C)$ be the larger of its numbers of forward and backward arcs. s-star discrepancy conjecture. If $s^*(D)\ge 0$, then there is a Hamilton oriented cycle $C$ in $D$ such that

$$\sigma_{\max}(C)\ge \left\lceil\frac{n+s^*(D)}{2}\right\rceil.$$

This is stated as another natural generalization of the cited Dirac-type theorem. The source presents it as open and reports supporting results for the related conjectures.