Bang-Jensen et al.'s dijoin conjecture for inversion number

For digraphs without loops, parallel edges, or $2$-cycles, let $L$ and $R$ be oriented graphs. The dijoin $L\Rightarrow R$ is formed from their disjoint union by adding an arc $uv$ for every $u\in V(L)$ and $v\in V(R)$. The inversion number $\operatorname{inv}(D)$ is the minimum size of a family of vertex subsets whose successive inversions make $D$ acyclic.

Bang-Jensen et al.'s dijoin conjecture. For oriented graphs $L$ and $R$,

$$\operatorname{inv}(L\Rightarrow R)=\operatorname{inv}(L)+\operatorname{inv}(R).$$

The conjecture is trivial when one of the two inversion numbers is zero and is known when their sum is at most $3$, as well as when both are $2$. It is false in general: counterexamples were independently found by Alon et al. and Aubian et al.