Variance-based anti-concentration conjecture for edge counts in Ramsey graphs

Let $C,c>0$ be fixed constants. Let $G$ be an $n$-vertex $C$-Ramsey graph, meaning that $G$ has no clique or independent set of size $C\log n$. Let $k$ satisfy

$$cn\le k\le (1-c)n,$$

and let $X$ be the number of edges induced by a uniformly random $k$-vertex subset of $G$.

Variance-based anti-concentration conjecture. For every $x\in\mathbb{Z}$,

$$\Pr(X=x)=O\mathopen{}\mathclose\bgroup\originalleft(\frac{1}{\sqrt{\operatorname{Var} X}}\aftergroup\egroup\originalright).$$

This would give an optimal point-probability bound in terms of the variance, as suggested by Chebyshev's inequality. The statement is presented as an interesting open direction for Ramsey graphs.