Conjecture on fixing numbers of symmetric groups

Let $S_n$ be the symmetric group on $n$ letters, and let $\operatorname{fix}(S_n)$ denote the set of possible numbers of vertices that fix a graph whose automorphism group is $S_n$. Fixing-number conjecture.

$$\operatorname{fix}(S_n)=\{1,\ldots,n-1\}.$$

The conjecture is motivated by the exact values in the first four rows of the paper's table, while the subsequent rows only provide lower and upper bounds; its resolution is not specified here.