On Cameron's question about the triviality in primitive permutation groups of the stabilizer of two points that is normal in the stabilizer of one of them

Assume that $G$ is a primitive permutation group on a finite set $X$, $x\in X$, $y\in X\setminus\{x\}$, and $G_{x, y} \trianglelefteq G_x$. P. Cameron raised the question about the validity of the equality $G_{x, y} = 1$ in this case. The author proved earlier that, if the socle of $G$ is not a direct power of an exceptional group of Lie type distinct from $E_6(q)$, $^2E_6(q)$, $E_7(q)$ and $E_8(q)$, then $G_{x, y} = 1$. In the present paper, we prove this in the case when the socle of $G$ is a direct power of an exceptional group of Lie type isomorphic to $E_6(q)$, $^2E_6(q)$, or $E_7(q)$.