Mixed-graph pseudo-dicut orientation conjecture

Let $M=(V,E\cup A)$ be a mixed graph, where $A$ is the set of directed arcs and $E$ the set of undirected edges. For $U\subset V$, a pseudo dicut is the edge set $\delta_E(U)$ such that no directed arc enters $U$. An orientation $E^+$ of $E$ replaces every undirected edge by a directed edge. For an oriented cut, write $\delta_{E^+}^+(U)$ and $\delta_{E^+}^-(U)$ for its outgoing and incoming edges.

Mixed-graph orientation conjecture. There is a constant $k>2$ such that for every mixed graph $M=(V,E\cup A)$, there is an orientation $E^+$ such that

$$|\delta_{E^+}^+(U)|,|\delta_{E^+}^-(U)|\geq \left\lfloor\frac{1}{k}|\delta_E(U)|\right\rfloor$$

for every pseudo dicut $\delta_E(U)$.

This is proposed as an analogue of nowhere-zero flow methods for mixed graphs. The source notes that the counterexample to Edmonds–Giles implies $k>2$, and gives no resolution.