Mixed-graph pseudo-dicut orientation threshold conjecture

Let $M=(V,E\cup A)$ be a mixed graph, with directed arcs $A$ and undirected edges $E$. For $U\subset V$, a pseudo dicut is $\delta_E(U)$ when no directed arc enters $U$. Its minimum pseudo dicut size is the smallest cardinality of a pseudo dicut.

Mixed-graph threshold orientation conjecture. There is a constant $k>2$ such that every mixed graph with minimum pseudo dicut size at least $k$ has an orientation $E^+$ satisfying

$$|\delta_{E^+}^+(U)|,|\delta_{E^+}^-(U)|\geq 1$$

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

This is presented as a weaker question than the preceding mixed-graph orientation conjecture; the source gives no resolution.