Does there exists a universal constant $C \geq 1$ such that if $X$ is any a smooth, toric, Fano $n$-dimensional manifold admitting a Kähler-Einstein metric, then its anticanonical degree $(-K_X)^n$ is at most $C^n (n+1)^n$?
This would follow from the weakening of Ehrhart's conjecture that I proposed in Reference request: Ehrhart's conjecture on the geometry of numbers (or at least this is what I understand from reading page 6 of the paper of Nill and Paffenholtz http://front.math.ucdavis.edu/0905.2054).
1. Is this known?
2. Is this interesting?
My knowledge of algebraic geometry is pitiful so please do not be offended if the question is really naive. I'm just trying to see what possible interesting consequences the weakened conjecture could have.