Sublinear-rank motion conjecture for primitive coherent configurations

Fix $0<\delta<1$. Let $\mathfrak{X}$ be a primitive coherent configuration on $n$ vertices, and suppose every constituent has degree at most $\delta n$. Sublinear-rank motion conjecture. Then

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

For bounded rank, the stronger linear bound follows from the paper's proposition. The conjecture concerns unbounded rank, where that argument does not apply; the stated $\Omega(n/\log(n))$ bound remains open.