Density of small diameter subgraphs in $K_r$-free graphs

We denote by $\text{ex}(n,H,F)$ the maximum number of copies of $H$ in an $n$-vertex graph that does not contain $F$ as a subgraph. Recently, Grzesik, Győri, Salia, Tompkins considered conditions on $H$ under which $\text{ex}(n,H,K_r)$ is asymptotically attained at a blow-up of $K_{r-1}$, and proposed a conjecture. In this note we disprove their conjecture.