The bounded short-arc lattice-point conjecture

For an integer $n$, count integer points on the circle $a^2+b^2=n$ whose second coordinate lies in a short interval. The short-arc lattice-point conjecture. For every $\alpha<1/2$, there is a constant $C_\alpha$ such that, for every $N$,

$$\#\{(a,b):a^2+b^2=n,\ N\le |b|<N+n^\alpha\}\le C_\alpha.$$

The source says this is proved for $\alpha\le1/4$ but remains open for $\alpha>1/4$.