Threshold existence conjecture for graph sequences

Let ${\cal G}=\{G_1,\ldots,G_n,\ldots\}$ be a graph sequence, and let $th({\cal G})$ denote its threshold when one exists. Threshold existence conjecture. Every graph sequence ${\cal G}$ has a threshold $th({\cal G})$. The conjecture concerns the missing general existence theorem for pebbling thresholds; the supplied status evidence says this implication is false.