Flag-transitive point-primitive quasi-symmetric $2$-designs and exceptional groups of Lie type

Let $\mathcal{D}$ be a non-trivial quasi-symmetric $2$-design with two block intersection numbers $x=0$ and $2\le y\le 10$, and suppose that $G$ is an automorphism group of $\mathcal{D}$. If $G$ is flag-transitive and point-primitive, then it is known that $G$ is either of affine type or almost simple type. In this paper, we show that the socle of $G$ cannot be a finite simple exceptional group of Lie type.