Aubian et al.'s universal counterexample conjecture for dijoin inversion

Let $L$ and $R$ be oriented graphs, let $L\Rightarrow R$ denote their dijoin, and let $\operatorname{inv}(D)$ denote the inversion number of an oriented graph $D$.

Aubian et al.'s conjecture. For every integer $k\ge 3$, there exists a tournament $T_k$ with

$$\operatorname{inv}(T_k)=k$$

such that, for every oriented graph $R$ with $\operatorname{inv}(R)\ge 1$,

$$\operatorname{inv}(T_k\Rightarrow R)<k+\operatorname{inv}(R).$$

The claim extends Aubian et al.'s proved construction for odd $k\ge3$ to all $k\ge3$; the supplied text gives no resolution of the even case.