Large deviation rate function for the upper tale of subgraphs count in Erdos-Renyi graphs

Question

Let $H_1,\cdots,H_k$ be $k$ fixed subgraphs on $r_1, \cdots, r_k$ vertices, respectively. Let $T_{1,n}, \cdots , T_{k,n}$ be the number of non-induced copies of $H_1,\cdots,H_k$ in an Erdos-Renyi graph $G(n,\frac{1}{2})$, respectively. I am interested in the upper tale problem $$\lim_{n\rightarrow \infty} \frac{1}{n^2}\log\mathbb{P}\left(T_{1,n}\geq t_1n^{r_1},\cdots, T_{k,n}\geq t_k n^{r_k}\right)$$ where $t_1, \cdots, t_k$ are parameters in $[0,1]$. In the case $k=1$ I am familiar with Chaterjee and Varadhan's work (section 4) using the framework of graphons. I am wondering if there is similar works that deal with cases where $k>1$?