The lower bound for negative oriented distinguishing index

Let $G$ be a connected graph. Here $D'(G)$ denotes the distinguishing index of $G$, and $OD'^-(G)$ denotes the negative oriented distinguishing index.

Lower-bound conjecture.

$$OD'^-(G)\geq \lfloor D'(G)/2\rfloor.$$

If true, this would show that a suitable orientation cannot reduce the number of required colours by more than half; in particular, every graph with a rigid orientation would have distinguishing index at most three. The conjecture is open.