The finite sparse-forcing conjecture

Let $\mathcal{S}$ be a set of graphs. It is sparse forcing if, whenever graphs $G_n$ satisfy $|G_n|\to\infty$, have edge density $p_n=|G_n|^{-o(1)}$, and the limits

$$c_F=\lim_{n\to\infty}t_{p_n}(F,G_n)$$

exist for every graph $F$ with $\sup_F c_F^{1/e_F}<\infty$, the condition $c_F=1$ for every $F\in\mathcal{S}$ implies $c_F=1$ for every graph $F$. Finite sparse-forcing conjecture. No finite set of graphs $\mathcal{S}$ can be sparse forcing. The paper's counterexample shows that no set of triangle-free graphs is sparse forcing, while this conjecture asserts the stronger impossibility for every finite set.