Alikhani–Soltani conjecture on distinguishing parameters of Mycielskian graphs

Let $G$ be a connected graph of order $n\geq 3$, and let $\mu(G)$ denote its Mycielskian graph. Write $\operatorname{Dist}(G)$ for the distinguishing number and $\operatorname{Dist^{\prime}}(G)$ for the distinguishing index. Alikhani–Soltani conjecture. Except for a finite number of graphs,

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

and

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

This conjecture predicts that passing to the Mycielskian does not increase either distinguishing parameter for all but finitely many connected graphs. The cited source proposes the claim; no resolution is given here.