The extremal-family conjecture for cliques in graphs with bounded minimum degree

Let $n$ and $\beta$ be such that the family of graphs $\mathcal{G}(n,\beta)$ is defined, and let $(n,\beta)$ be feasible when $G[V_0]$ is triangle-free for $G\in\mathcal{G}(n,\beta)$. Write $\delta=(1-\beta)n$, let $p$ be the parameter appearing in the definition of $\mathcal{G}(n,\beta)$, and let $g_r(\beta)$ denote the associated clique-density expression. The quantity $k_r(n,\delta)$ is the minimum possible number of $r$-cliques in an $n$-vertex graph with minimum degree at least $\delta$. The extremal-family conjecture. Let $n$ and $\delta$ be positive integers. Then

$$k_r(n,\delta)\ge g_r(\beta)n^r,$$

where $\delta=(1-\beta)n$ and $r\ge3$. Moreover, for $3\le r\le p+1=\lceil\beta^{-1}\rceil$, equality holds if and only if $(n,\beta)$ is feasible and the extremal graphs are members of $\mathcal{G}(n,\beta)$. This extends the known Turán-theoretic cases; the conjecture is true when $p=1$ or $r>p+1$, and also when $\beta=1/(p+1)$ with $(p+1)\mid n$, while the stated range in general remains open.