The full range of counterexamples to the dijoin conjecture

Let $L$ and $R$ be oriented graphs, and let $L\rightarrow R$ be their dijoin. The dijoin conjecture asserts that its inversion number equals the sum of the inversion numbers of the factors.

Dijoin counterexample-range conjecture. For all $\ell,r\in\mathbb{N}$ with $\ell\geq 3$ or $r\geq 3$, there exist oriented graphs $L$ and $R$ such that

$$\operatorname{inv}(L)=\ell,\qquad \operatorname{inv}(R)=r,$$

and

$$\operatorname{inv}(L\rightarrow R)<\ell+r.$$

The paper has counterexamples with one inversion number equal to $3$ and explains an equivalent formulation using tournaments, but the general assertion remains conjectural.