Motion conjecture for primitive coherent configurations

Let $\mathfrak X$ be a primitive coherent configuration (PCC) with $n$ vertices. Its motion is the minimum number of vertices moved by a non-identity automorphism of the configuration. A Cameron scheme is one of the exceptional primitive coherent configurations named in the source.

Motion conjecture for primitive coherent configurations. If $\mathfrak X$ is not a Cameron scheme, then there is a positive constant $c$ such that

$$\operatorname{motion}(\mathfrak X)\geq cn.$$

The conjecture is motivated by the Liebeck--Saxl result proving the analogous statement with $c=1/3$ in the Schurian case. The general PCC case is presented as an open direction motivating subsequent work.