Threshold-range conjecture for graph sequences

Let $t_1$ and $t_2$ be functions such that $t_1\in\Omega(n^{1/2})$, $t_2\in O(n)$, and $t_1\ll t_2$. Threshold-range conjecture. There is a graph sequence ${\cal G}=\{G_1,\ldots,G_n,\ldots\}$ such that

$$th({\cal G})\in\Omega(t_1)\cap O(t_2).$$

Moreover, there is such a sequence in which each $G_n$ is a tree. This conjecture asks whether threshold functions for graph sequences can realize a broad range between the known lower and upper scales; the supplied status evidence says the implication is false.