Lower-bound conjecture for inversion number of tournament dijoins

Let $D_1$ and $D_2$ be tournaments. Let $\operatorname{inv}(D)$ be the inversion number and let $\operatorname{tmr}(D)$ be the tournament minimum rank.

Lower-bound conjecture for tournament dijoins. One should have

$$\operatorname{inv}(D_1\rightarrow D_2)\geq\operatorname{inv}(D_1)+\operatorname{inv}(D_2)-1,$$

with equality if and only if

$$\operatorname{inv}(D_1)=\operatorname{tmr}(D_1)+1$$

or

$$\operatorname{inv}(D_2)=\operatorname{tmr}(D_2)+1.$$

The conjecture would improve the lower bound suggested by the proposed tournament-minimum-rank additivity statement. The supplied text gives no resolution, so the claim remains open.