Bérczi–Chandrasekaran conjecture on head-disjoint hypergraph orientations

Let $H=(V,\mathcal E)$ be a $\tau$-uniform hypergraph. For $X\subset V$, let $d_H(X)$ be the sum of $|X\cap E|$ over all hyperedges $E\in\mathcal E$ separated by $X$, meaning $E\cap X\neq\emptyset$ and $E\not\subseteq X$. A strongly connected orientation of $H$ designates one vertex of each hyperedge as its head, with every nonempty $X\subset V$ having a hyperedge whose head lies in $X$ and which also contains a vertex outside $X$. Two orientations are head-disjoint when no hyperedge has the same head in both orientations. Bérczi–Chandrasekaran's conjecture. If $d_H(X)\geq\tau$ for every nonempty $X\subset V$, then $H$ has $\tau$ pairwise head-disjoint strongly connected orientations. This is presented as an unpublished conjecture extending the graph case to general uniform hypergraphs; the supplied text gives no resolution.