The conjecture on negative oriented distinguishing index when the distinguishing index is two

Let $G$ be a connected graph, and let $D'(G)$ denote its distinguishing index and $OD'^-(G)$ its negative oriented distinguishing index.

Conjecture for distinguishing index two. If

$$D'(G)=2,$$

then

$$OD'^-(G)=1.$$

This is motivated by the preceding determination of the parameter for several graph classes and remains open. It concerns the values of the negative oriented distinguishing index for graphs whose distinguishing index equals two.