The influential-pair obstruction conjecture for monochromatic subgraph counts

Let $H$ be a fixed graph and let ${G_n}$ be a sequence of graphs. For vertices $v,w$ of $G_n$, write ${\mathbf{w}}=(v,w)\in V(G_n)_2$ for a pair of vertices, and call such a pair influential when it has the influential-pair property considered in the paper. Let $Z(H,G_n)$ denote the normalized monochromatic $H$-count.

Influential-pair obstruction conjecture. If for every $N$ there is some $n>N$ such that $G_n$ has an influential pair of vertices ${\mathbf{w}}=(v,w)\in V(G_n)_2$, then

$$Z(H,G_n)\centernot{\xrightarrow{\text{in law}}}\mathscr{N}(0,1).$$

This proposes that influential pairs prevent convergence in law to the standard normal distribution for general monochromatic subgraph counts. The paper proves a sufficiency result under the absence of influential pairs, while the corresponding necessity statement is conjectured because the quadratic-polynomial argument available for triangles does not directly extend to general graphs $H$.