The direction-independence conjecture for dijoin inversion number

Let $L$ and $R$ be oriented graphs, and let $L\Rightarrow R$ be the dijoin obtained by directing every cross-edge from $L$ to $R$. Let $\operatorname{inv}(D)$ denote the inversion number.

Direction-independence conjecture. For every oriented graphs $L$ and $R$,

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

The equality is proved in the special case involving the directed $3$-cycle in one direction, but the source describes the general assertion as non-trivial and conjectural.