Mycielskian symmetry-breaking conjecture

Let $G$ be a connected graph of order $n\geq 3$, and let $\mu(G)$ denote its Mycielskian. Write $D(G)$ for the distinguishing number of $G$ and $D'(G)$ for its distinguishing index. Mycielskian symmetry-breaking conjecture. Except for a finite number of graphs, one has

$$D(\mu(G))\leq D(G)\quad\text{and}\quad D'(\mu(G))\leq D'(G).$$

The conjecture proposes that applying the Mycielskian does not increase either the distinguishing number or the distinguishing index, apart from finitely many connected graphs. The preceding results establish this monotonicity for the sequence of Mycielski graphs from $M_i$ to $M_{i-1}$ for $i\geq 5$, but the general assertion remains open.