The conjecture that the equal-weight subgraph-count property is quasi-random

Let $F$ be a graph with $e(F)>1$, and let $m=|F|$. For a graph property $\mathcal P(F;1/m,\dots,1/m)$ and its weaker version \widetilde\mathcal P(F;1/m,\dots,1/m), the source context defines these as properties concerning equal-weight subgraph counts. Equal-weight subgraph-count conjecture. Theorem~ holds for any graph $F$ with $e(F)>1$; equivalently, $\mathcal P(F;1/m,\dots,1/m)$ and \widetilde\mathcal P(F;1/m,\dots,1/m) are quasi-random properties for every such graph $F$. The conjecture extends the theorem proved for regular graphs, stars, and disconnected graphs; the only indicated counterexample is $F=K_2$, which has $e(F)=1$.