Bang-Jensen et al.'s NP-completeness conjecture for bounded inversion number

For a digraph $D$, let $\operatorname{inv}(D)$ be its inversion number, and let $k$ be a fixed positive integer.

Bang-Jensen et al.'s complexity conjecture. Deciding whether a given digraph $D$ has inversion number at most $k$ is NP-complete for any fixed positive integer $k$.

The paper notes that the case $k=1$ is already known to be NP-complete and that the dijoin conjecture would imply the asserted result for every fixed $k$. The source does not state a resolution.