Partition conjecture for strengthening sets of a crossing family

Let $D=(V,A)$ be a weakly connected digraph, let $\mathcal{C}$ be a crossing family over $V$, and let a strengthening set of $D$ for $\mathcal{C}$ be a subset $J\subseteq A$ such that every set in $\mathcal{C}$ has at least one outgoing arc after the arcs in $J$ are flipped. Let $\tau\geq2$ be an integer.

Strengthening-set partition conjecture. If

$$|\delta_D^-(U)|+(\tau-1)|\delta_D^+(U)|\geq\tau$$

for every $U\in\mathcal{C}$, then $A$ can be partitioned into $\tau$ strengthening sets of $D$ for $\mathcal{C}$.

The conjecture unifies several directed-cut partition statements: for $\tau=2$ it is equivalent to the paper's strong-orientation theorem, and a suitable choice of $\mathcal{C}$ yields Woodall's conjecture. Its validity in the general crossing-family setting remains open.