Babai's motion conjecture for primitive distance-regular graphs

Let $d\geq 3$, and let $X$ be a primitive distance-regular graph of diameter $d$ on $n$ vertices. Write $\operatorname{motion}(X)$ for the minimum number of vertices moved by a nonidentity automorphism of $X$.

Babai's motion conjecture. There exists an $\alpha>0$ such that either

$$\operatorname{motion}(X)\geq \alpha n,$$

or $X$ is a Hamming graph, or a Johnson graph.

This is the specialization of Babai's coherent-configuration conjecture to primitive distance-regular graphs. The paper discusses partial results, including the rank-$3$ and rank-$4$ cases, but the general statement remains open.