On transitive sets of derangements in primitive groups

We construct a primitive permutation action of the Steinberg triality group $^3D_4(2)$ of degree $4064256$ and show that there are distinct points $\alpha,\beta$ such that there is no derangement $g\in{^3D_4}(2)$ with $\alpha^g=\beta$. This answers a question by John G. Thompson (Problem 8.75 in the Kourovka Notebook) in the negative.