Conjecture on lattice points of circles in short horizontal intervals

Let $n$ be a positive integer, let $\alpha<1/2$, and let $N$ be any number. Consider the integer lattice points $(a,b)$ on the circle

$$a^2+b^2=n$$

whose second coordinate satisfies

$$N\leq |b|<N+n^\alpha.$$

Circle lattice-point conjecture. There is a constant $C_\alpha$ such that, for every $N$, the number of these lattice points is at most $C_\alpha$.

The conjecture concerns uniformly bounded representations of a fixed circle by lattice points in short intervals near the horizontal axis. The source attributes it externally and gives no indication that it has been resolved.