Anticoncentration conjecture for subgraph counts in random graphs

Let $G\sim G(n,p)$, where $p\in(0,1)$ is fixed, and let $X_H$ denote the number of copies of a fixed graph $H$ in $G$. Assume that $H$ has no isolated vertices, and write $v(H)$ for its number of vertices.

Anticoncentration conjecture.

$$\max_{x\in\mathbb{N}}\Pr(X_H=x)=O\mathopen{}\mathclose\bgroup\originalleft(n^{1-v(H)}\aftergroup\egroup\originalright).$$

This conjecture proposes the optimal-order uniform upper bound for point probabilities of subgraph counts. Earlier work gives the weaker bound $n^{-1+o(1)}$ for graphs containing an edge; the conjectured estimate remains the sharper target in general.