A vertex whose deletion decreases inversion number by at most one

Let $D$ be a digraph with at least one vertex. For $v\in V(D)$, write $D-\{v\}$ for the digraph obtained by deleting $v$ and all incident edges.

Vertex-deletion conjecture. There exists $v\in V(D)$ such that

$$\operatorname{inv}(D-\{v\})\geq\operatorname{inv}(D)-1.$$

Equivalently, the general bound $\operatorname{inv}(D)\leq\operatorname{inv}(D-\{v\})+2$ cannot be tight for every vertex of the same digraph. The supplied text gives no resolution.