Cilleruelo–Granville arc conjecture for lattice points on semicircles

For $N\in\mathbb{N}$, let $\mathbb{S}^1_+(N)$ be the upper semicircle of radius $\sqrt N$, and let $S_N=\mathbb{S}^1_+(N)\cap\mathbb{Z}^2$. Cilleruelo–Granville conjecture. For every $\gamma\in(0,1)$, every arc in $\mathbb{S}^1_+(N)$ of length $N^{\gamma/2}$ contains at most $C(\gamma)$ lattice points.

The conjecture is used in the paper through a weaker form to remove an $N^\epsilon$ loss from a fourth-moment estimate; the source states that the conjecture was proved for all $\gamma<\frac12$ in the cited work [CC], so the full assertion remains open.