Flag-transitive automorphism groups of $2$-designs with $\lambda\geq (r,\lambda)^2$ are not product type

In this paper we show that a flag-transitive automorphism group $G$ of a non-trivial $2$-$(v,k,\lambda)$ design with $\lambda\geq (r, \lambda)^2$ is not of product action type. In conclusion, $G$ is a primitive group of affine or almost simple type.