Strong Ramadanov conjecture for strictly pseudoconvex domains

Let $D$ be a bounded strictly pseudoconvex domain in $\mathbb{C}^n$ with $C^\infty$ boundary. Let $\rho$ be its Fefferman potential function, namely the solution of the Fefferman equation described in the paper.

Strong Ramadanov conjecture. If $\rho \in C^{n+2}(\overline{D})$, then $\partial D$ is locally spherical.

The conjecture characterizes the vanishing of the obstruction function in the Fefferman expansion: the stated boundary regularity is expected to force local CR-equivalence of the boundary to a sphere. The paper proves this conjecture for real ellipsoids in $\mathbb{C}^2$ by two different approaches.