The distinguishing stability bound for simple connected graphs

Let $G$ be a simple connected graph. Write $D(G)$ for its distinguishing number and $st_D(G)$ for its distinguishing stability, namely the least number of vertices whose deletion changes the distinguishing number.

Distinguishing stability conjecture.

$$st_D(G)\leqslant D(G)+1.$$

The authors state this bound because it holds for the examples they have considered, but report that attempts to prove it have failed. Its general validity remains open.