Essential norm estimates for Hankel operators on convex domains in $\mathbb{C}^2$

Let $\Omega\subset \mathbb{C}^2$ be a bounded convex domain with $C^1$-smooth boundary and $\varphi\in C^1(\overline{\Omega})$ such that $\varphi$ is harmonic on the nontrivial disks in the boundary. We estimate the essential norm of the Hankel operator $H_{\varphi}$ in terms of the $\overline{\partial}$ derivatives of $\varphi$ "along" the nontrivial disks in the boundary.