Transitive sets of derangements in primitive actions of PSL_2(q)

Problem 8.75 of the Kourovka Notebook [10], attributed to John G. Thompson, asks the following: Suppose $G$ is a finite primitive permutation group on $\mathrm{\Omega }$, and $\alpha$, $\beta$ are distinct points of $\mathrm{\Omega }$. Does there exist an element $g\in G$ such that ${\alpha }^g=\beta$ and $g$ fixes no point of $\mathrm{\Omega }$? A recent negative example is given in [12], where $G$ is the Steinberg triality group ${}^3{D}_4(2)$ acting primitively on 4,064,256 points. At present this is the only negative example known. In this note we show that almost simple primitive permutation groups with socle isomorphic to PSL_2(q) do not give negative examples.