Schmidt's conjecture on integral points on smooth strictly convex curves

Let $N>0$, let $I$ be the interval under consideration, and let $f:[0,N]\to[0,N]$ be a strictly convex function such that $f\in C^3([0,N])$ and $f^{(3)}(x)\neq 0$ for all $x\in[0,N]$. Schmidt's conjecture. The number of integral points on the graph of $f$ above $I$ satisfies

$$\left\lvert\{(x,f(x)):x\in I\}\cap\mathbb{Z}^2\right\rvert\ll N^{\frac12+o(1)}.$$

This conjecture predicts that the exponent $3/5$ in Schmidt's uniform bound can be improved to $1/2$ under the same $C^3$ and nonvanishing third-derivative assumptions. It concerns the sharp order of magnitude for integral points on smooth strictly convex curves.