Block-transitive $t$-($k^2,k,\lambda$) designs with $PSL(n,q)$ as socle

Let $\mathcal{D}=(\mathcal{P},\mathcal{B})$ be a non-trivial block-transitive $t$-$(k^2,k,\lambda )$ design with $G\le \text{Aut}(\mathcal{D})$ and $X⊴G\le \text{Aut}(X)$, where $X=PSL(n,q)(n\ge 3)\mathrm{.}$ We prove that $t=2$ and the parameters $(n,q,v,k)$ is $(3,3,144,12),(4,7,400,20)$ or $(5,3,121,11)\mathrm{.}$ Moreover, $\mathcal{D}$ is a $2$-$(144,12,\lambda )$ design with $\lambda \in \{3,6,12\}$ if $\lambda \mid k$.