Lattice points in large convex planar domains of finite type

Let $\mathcal{B}$ be a compact convex planar domain with smooth boundary of finite type and $\mathcal{B}_\theta$ its rotation by an angle $\theta$. We prove that for almost every $\theta\in[0, 2\pi]$ the remainder $P_{\mathcal{B}_\theta}(t)$ is of order $O_{\theta}(t^{2/3-\zeta})$ with a positive number $\zeta$ independent of the domain.