Weaker variant of Woodall's dijoin decomposition conjecture

Let $D=(V,A)$ be a digraph, and let a dijoin be a set of arcs meeting every dicut. A set of arcs is a $k$-dijoin if it can be decomposed into $k$ dijoins. Let $\tau\geq 2$ be an integer, and suppose every dicut of $D$ has size at least $\tau$.

Weaker dijoin decomposition conjecture. The arc set $A$ can be decomposed into a $k$-dijoin and a $(\tau-k)$-dijoin, for every $k\in\{1,\ldots,\tau-1\}$.

This is presented as a weaker variant of Woodall's conjecture, which asserts that the minimum size of a dicut equals the maximum number of pairwise disjoint dijoins and remains open. The case $k=1$ and $k=\tau-1$ is known, while the intermediate cases remain open.