Belkhechine et al.'s lower-bound conjecture for the inversion number

For every positive integer $n$, let

$$\operatorname{inv}(n)=\max\{\operatorname{inv}(D)\mid D\text{ is an oriented graph of order }n\}.$$

Here $\operatorname{inv}(D)$ is the minimum number of inversions needed to transform $D$ into an acyclic oriented graph.

Belkhechine et al.'s inversion-number conjecture.

$$\operatorname{inv}(n)\geq\left\lfloor\frac{n-1}{2}\right\rfloor.$$

This strengthens the elementary counting lower bound given immediately before it and asserts that the logarithmic loss in that bound is unnecessary. The source does not state a resolution.