A case of the dijoin conjecture on inverting oriented graphs

For an oriented graph $D$, the inversion of $X\subseteq V(D)$ in $D$ is the graph obtained by reversing the orientation of all arcs with both ends in $X$. The inversion number $\mathrm{i}\mathrm{n}\mathrm{v}(D)$ is the minimum number of inversions needed to obtain an acyclic oriented graph. We show that the dijoin conjecture of Bang-Jensen, da Silva and Havet, that $\mathrm{i}\mathrm{n}\mathrm{v}(D_1\to D_2)=\mathrm{i}\mathrm{n}\mathrm{v}(D_1)+\mathrm{i}\mathrm{n}\mathrm{v}(D_2)$, is true in the case where $\mathrm{i}\mathrm{n}\mathrm{v}(D_1)=2$ and $\mathrm{i}\mathrm{n}\mathrm{v}(D_2)$ is even. We also characterise the cases $\mathrm{i}\mathrm{n}\mathrm{v}(D_1)=2$ and $\mathrm{i}\mathrm{n}\mathrm{v}(D_2)$ odd, for which the conjecture does and does not hold. We then go on to show a similar result for n-joins, in doing so we prove a conjecture of Alon, Powierski, Savery, Scott and Wilmer. Our proofs build on the idea of tournament minimum rank, introduced by Behague, Johnston, Morrison and Ogden.