Uniform anti-concentration conjecture for the independence number of sparse random graphs

Let $G_{n,p}$ be the binomial random graph with $n$ vertices, where $p=n^{\gamma}$ and $-1<\gamma<-2/3$, and let $\alpha(G_{n,p})$ denote its independence number. For an arbitrary sequence $k=k(n)$, Uniform anti-concentration conjecture.

$$\mathbb{P}\left(\alpha(G_{n,p})=k\right)=\widetilde{O}\left(n^{1+3\gamma/2}\right).$$

This is proposed as a strengthening of the cited theorem and would give a uniform upper bound on every point probability in the regime $-1<\gamma<-2/3$; the supplied text gives no evidence that it has been resolved.