The bounded lattice-points-in-diagonal-arcs conjecture

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

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

The diagonal-arc conjecture. The number of these lattice points in an arc of length $R^{1-\epsilon}$ around the diagonal is bounded uniformly in $R$. The source presents this as the special case corresponding to a short interval near the diagonal; it remains open in the supplied text.