Strong spanning subdigraphs in highly arc-strong and arc-antistrong digraphs

Let $D$ be a digraph. It is $k$-arc-strong when deleting fewer than $k$ arcs leaves a strongly connected digraph, and $k$-arc-antistrong when deleting fewer than $k$ arcs leaves an antistrong digraph.

Special-case conjecture. There exists a natural number $k$ such that every digraph $D$ which is both $k$-arc-strong and $k$-arc-antistrong has arc-disjoint strong spanning subdigraphs $D_1,D_2$.

This is proposed as a potentially easier special case of the preceding connectivity conjecture. The supplied text gives no resolution, so its status is open.