Dijoin conjecture for joins of 2-invertible oriented graphs

Let $k\in\mathbb{N}$ and let $D_1,\dots,D_k$ be oriented graphs satisfying

$$\operatorname{inv}(D_i)\leq 2\qquad\text{for all }i.$$

Write $[D_1,\dots,D_k]$ for their join, and let $\operatorname{inv}$ denote inversion number. Dijoin conjecture. Then

$$\operatorname{inv}([D_1,\dots,D_k])=\sum_{i=1}^k\operatorname{inv}(D_i).$$

This conjecture is refuted: a counterexample exists with $\operatorname{inv}(L)=1$ and $\operatorname{inv}(R)=3$, so the asserted equality does not hold in general.