Extremal Kaehler-Einstein metric for two-dimensional convex bodies

Given a convex body $K\subset {\mathbb{R}}^n$ with the barycenter at the origin we consider the corresponding K{"a}hler-Einstein equation $e^{-\Phi} = \det D^2 \Phi$\mathrm{.}If$K$ is a simplex, then the Ricci tensor of the Hessian metric $D^2\mathrm{\Phi }$ is constant and equals $\frac{n-1}{4(n+1)}$. We conjecture that the Ricci tensor of $D^2\mathrm{\Phi }$ for arbitrary $K$ is uniformly bounded by $\frac{n-1}{4(n+1)}$ and verify this conjecture in the two-dimensional case. The general case remains open.