Barycentric transformation conjecture for smooth Fano polytopes

Let $P$ be a smooth Fano polytope, and let $B(P)$ denote its barycentric transformation. A Fano polytope is of type $B_\infty$ when the barycentric transformation can be applied indefinitely. Barycentric transformation conjecture.

1. If $P$ has odd dimension, then $P$ is Kähler–Einstein if and only if $P$ is of type $B_\infty$.
2. If $P$ is Kähler–Einstein, then $B(P)$ is also a Kähler–Einstein Fano polytope. In particular, $P$ is of type $B_\infty$.
3. If $P$ is symmetric, then $B(P)$ is a symmetric and Kähler–Einstein Fano polytope. In particular, $P$ is of type $B_\infty$.

The conjecture extends the authors' low-dimensional results on preservation of the Kähler–Einstein property under barycentric transformations. The even-dimensional converse related to part 2 is discussed separately, and the validity for singular Fano polytopes is posed as an additional question.