Cilleruelo–Granville conjecture on lattice points in short arcs

Let $m$ be a positive integer and consider the circle of radius $R=\sqrt{m}$. An arc is a connected portion of this circle, and lattice points are points of $\mathbb{Z}^2$ lying on the circle.

Cilleruelo–Granville conjecture. For every $\delta>0$, there exists a constant $C_{\delta}$ such that every arc of length $R^{1-\delta}$ contains at most $C_{\delta}$ lattice points.

This conjecture concerns uniform bounds for lattice points on arcs whose length is a power strictly below the radius. The supplied text presents it as part of the known and conjectural landscape for short arcs, but gives no resolution.