Amended Pikhurko–Razborov conjecture on extremal clique-minimizing graphs

Fix $r\ge4$. For positive integers $n,e$ with $e\le\binom{n}{2}$, let $t_{r-1}(n)$ be the number of edges in the balanced complete $(r-1)$-partite graph, let $g_r(n,e)$ be the minimum number of copies of $K_r$ in an $(n,e)$-graph, and let $\mathcal{H}_1^{\ast}(n,e)$ and $\mathcal{H}_2^{\ast}(n,e)$ be the two explicitly defined graph families in the source. Amended Pikhurko–Razborov conjecture. For every sufficiently large integer $n$ and every integer $e$ satisfying $t_{r-1}(n)<e\le\binom{n}{2}$,

$$\\{G:G\text{ is an }(n,e)\text{-graph with }N(K_r,G)=g_r(n,e)\\}=\mathcal{H}_1^{\ast}(n,e)\cup\mathcal{H}_2^{\ast}(n,e).$$

This is proposed after the original Pikhurko–Razborov strengthening is disproved; the proposition immediately preceding it verifies the asserted clique count for graphs in the two candidate families and shows that the second family is genuinely larger for infinitely many parameter pairs.