Xiao and Katona's clique-covering conjecture

Let $t_{k-1}(n)$ denote the maximum number of edges in a $K_{k-1}$-free graph on $n$ vertices, and let $\tau_k(G)$ be the minimum size of a vertex set meeting every copy of $K_k$ in $G$. For a balanced partition $V_1\cup\cdots\cup V_{k-1}$ of $[n]$, assume $|V_1|\geq\cdots\geq|V_{k-1}|$. Xiao and Katona's conjecture. For fixed integers $k\geq4$ (and as stated in the source, $s>t\geq1$), every graph $G$ on $n$ vertices with $t_{k-1}(n)+1$ edges and $\tau_k(G)\geq2$ contains at least

$$\left(|V_1|+|V_2|-2\right)\prod_{i=3}^{k-1}|V_i|$$

copies of $K_k$. The paper states that it gives a counterexample to one Xiao–Katona conjecture and proves a modified version, so this original conjecture is refuted.