The bounded-constituent-degree motion conjecture for coherent configurations

Fix $0<\delta<1$, and let $\mathfrak{X}$ be a primitive coherent configuration on $n$ vertices. For each constituent of $\mathfrak{X}$, let its degree be the common number of related vertices.

Motion conjecture under bounded constituent degrees. If every constituent has degree at most $\delta n$, then

$$\operatorname{motion}(\mathfrak{X})=\Omega\left(\frac{n}{\log n}\right).$$

For bounded rank, the proposition preceding this conjecture gives a linear lower bound. The asserted $\Omega(n/\log n)$ bound remains open when the rank is unbounded.