Klartag–Kolesnikov's Ricci curvature conjecture for Kähler–Einstein Hessian metrics

Let $K\subset\mathbb{R}^n$ be a convex body with barycenter at the origin, and let $\Phi$ solve the Kähler–Einstein equation

$$e^{-\Phi}=\det D^2\Phi.$$

For the simplex

$$S=\left\{(x_1,\ldots,x_n)\in\mathbb{R}^n\;\ \sum_{i=1}^n x_i\leq 1,\ \forall i,\ x_i\geq -1\right\},$$

the corresponding Hessian metric $D^2\Phi$ has Ricci tensor $\frac{n-1}{4(n+1)}D^2\Phi$. Klartag–Kolesnikov's Ricci curvature conjecture. The Ricci curvature of $D^2\Phi$ is bounded by $\frac{n-1}{4(n+1)}$, with the largest value realized uniformly on $S$. The existence and uniqueness of solutions to the equation are known under various assumptions, and the conjecture is motivated by the role of the equation in toric geometry and convex analysis. The authors verify the conjecture in dimension two, while its general validity remains open.