Counting $t$-wise $L$-intersecting cliques with prescribed intersection sizes

Let $r,t\ge 3$ be integers and $L=\{\ell_1,\ell_2,\ldots,\ell_s\}\subseteq [0,r-1]$ a fixed set of integers with $|L|\neq r$ and $\ell_1<\ell_2<\cdots<\ell_s$. For each integer $n$, let $Ψ_{r}(n,L,t)$ be the maximum number of $r$-cliques in an $n$-vertex graph whose $r$-cliques, viewed as a family o