Lidický–Murphy conjecture on generalized Turán numbers

Let $H$ be a graph, let $r$ be an integer with $r>\chi(H)$, and let $n$ be a positive integer. For positive integers $n_1,n_2,\dots,n_{r-1}$ satisfying

$$n_1+n_2+\dots+n_{r-1}=n,$$

Lidický–Murphy conjecture. There exist such integers for which

$$\operatorname{ex}(n,H,K_r)=H(K_{n_1,n_2,\dots,n_{r-1}}).$$

This conjectures that the maximum number of copies of $H$ in an $n$-vertex $K_r$-free graph is attained by a complete $(r-1)$-partite graph. The paper presents a counterexample to this conjecture, so the conjecture is refuted.