The bounded lattice-points-in-arcs conjecture for circles

For $R>0$, consider the lattice points on the circle

$$\{(x,y)\in\mathbb Z^2:x^2+y^2=R^2\}.$$

The circle-arc conjecture. The number of these lattice points in any arc of length $R^{1-\epsilon}$ is bounded uniformly in $R$. The source states the equivalent short-interval formulation for representations as sums of two squares and notes that the result is known only for shorter arcs.