The cycle extension distinguishing number conjecture

Let $C_n$ be the cycle graph on $n$ vertices, and let $\operatorname{ext}_D(C_n)$ denote the minimum number of blanks such that every coloring of the remaining vertices extends to a distinguishing coloring of $C_n$.

Cycle extension conjecture.

$$\operatorname{ext}_D(C_n)=\begin{cases}4&\text{if }n\not\equiv 0\pmod{4\text{ or }5},\\5&\text{if }n\equiv 0\pmod 4,\\6&\text{if }n\equiv 0\pmod 5. \end{cases}$$

The conjecture is sharp for the displayed cases, and the paper verifies it whenever the minimum prime divisor of $n$ is at least $7$; the general case remains open.