Optimal anti-concentration conjecture for subgraph counts in random graphs

Fix $p\in(0,1)$ and a graph $H$ with $h$ non-isolated vertices. Let $G\in\mathbb{G}(n,p)$ be a binomial random graph, and let $X_H$ denote the number of copies of $H$ in $G$. For any $x\in\mathbb{N}$, the optimal anti-concentration conjecture asserts

$$\Pr(X_H=x)=O\mathopen{}\mathclose\bgroup\originalleft(1/\sqrt{\operatorname{Var}(X_H)}\aftergroup\egroup\originalright)=O\mathopen{}\mathclose\bgroup\originalleft(1/n^{h-1}\aftergroup\egroup\originalright).$$

The paper states that the previously proved $O(1/n)$ bound is far from optimal, motivating this stronger conjectural estimate. The source does not provide a resolution or a formal name for the conjecture.