Lovász–Simonovits conjecture on minimum clique counts

For integers $n,e,r$, let $g_r(n,e)$ be the minimum number of copies of $K_r$ in an $n$-vertex graph with $e$ edges. Let $\mathcal{H}(n,e)=\mathcal{H}_1(n,e)\cup\mathcal{H}_2(n,e)$ be the prescribed family of multipartite $(n,e)$-graphs, and define

$$h_r(n,e):=\min\{K_r(H):H\in\mathcal{H}(n,e)\}.$$

Lovász–Simonovits conjecture. For every integer $r\geq 3$, there exists $n_0=n_0(r)>0$ such that

$$g_r(n,e)=h_r(n,e)$$

for all positive integers $n\geq n_0$ and $e\leq \binom{n}{2}$. This conjecture extends the Lovász–Simonovits theorem from a neighbourhood of the relevant Turán densities to the full range of edge counts. The source reports partial cases, including the theorem proved there for triangles away from the complete-graph density, but does not state a complete resolution.