Exact extremal-graph conjecture for the Lovász–Simonovits triangle problem

Let $g_3(n,e)$ be the minimum number of triangles in an $(n,e)$-graph. Let $\mathcal{H}_0^*(n,e)$ and $\mathcal{H}_2^*(n,e)$ denote the subclasses of $\mathcal{H}_0(n,e)$ and $\mathcal{H}_2(n,e)$ attaining the prescribed minimum $h^*(n,e)$. Exact extremal-graph conjecture. For all positive integers $n$ and $e\leq\binom{n}{2}$, an $(n,e)$-graph $G$ satisfies

$$K_3(G)=g_3(n,e)$$

if and only if

$$G\in\mathcal{H}_0^*(n,e)\cup\mathcal{H}_2^*(n,e).$$

This strengthens the triangle case of the Lovász–Simonovits conjecture by specifying every extremal graph, not only the minimum number of triangles. The source presents it as a belief and proves the equality of the minimum values asymptotically when the edge density is bounded away from $1$, but does not establish the claimed classification for all $n,e$.