Gishboliner–Krivelevich–Michaeli oriented discrepancy conjecture

Let $G$ be an oriented graph on $n\geq 3$ vertices, and let $\delta(G)$ denote its minimum degree. For a Hamilton cycle $C$ in $G$, let $\sigma_{\max}(C)$ be the larger of the numbers of forward and backward edges of $C$. Gishboliner–Krivelevich–Michaeli's conjecture. If

$$\delta(G)\geq \frac{n}{2},$$

then there exists a Hamilton cycle $C$ in $G$ such that

$$\sigma_{\max}(C)\geq \delta(G).$$

This conjecture is a directed analogue of discrepancy results for Hamilton cycles, predicting that a minimum-degree condition at least half the order forces a Hamilton cycle with sufficiently large oriented discrepancy. Its resolution is not indicated in the supplied text.