Alikhani–Soltani distinguishing-index conjecture for Mycielskians

Let $G$ be a connected graph with at least three vertices. Let $\operatorname{Dist}'(G)$ denote the distinguishing index of $G$, namely the least number of colors in a distinguishing edge coloring, and let $\operatorname{Dist}'(\mu(G))$ denote the distinguishing index of its Mycielskian.

Alikhani–Soltani's conjecture. For all but a finite number of connected graphs $G$ with at least three vertices,

$$\operatorname{Dist}'(\mu(G)) \leq \operatorname{Dist}'(G).$$

This conjecture predicts that the Mycielskian preserves the distinguishing index for all but finitely many connected graphs. The source gives an upper bound for twin-free graphs under additional hypotheses, but does not identify the finite exceptional family.