The all-$k$ counterexample conjecture for dijoin additivity

For an integer $k\geq3$, let $T_k$ be a tournament and let $R$ be an oriented graph; $T_k\Rightarrow R$ denotes their dijoin, formed by adding all arcs from $T_k$ to $R$.

The all-$k$ dijoin counterexample conjecture. For any $k\geq3$, there is a tournament $T_k$ with $\operatorname{inv}(T_k)=k$ such that

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

for all $R$ with $\operatorname{inv}(R)\geq1$.

The paper proves this statement for every odd integer $k\geq3$ and conjectures that the same phenomenon holds for even integers. Thus the remaining content concerns even $k$.