The Fano volume inequality conjecture

Let $Z$ be a smooth Fano manifold of complex dimension $n-1$, with Fano index $I(Z)mathrel{\in}\mathbb{Z}_{>0}$. Fano volume inequality conjecture.

$$I(Z)\int_Z c_1(Z)^{n-1}\leq n\int_{\mathbb{CP}^{n-1}}c_1(\mathbb{CP}^{n-1})^{n-1}=n^n.$$

Equality should hold if and only if $Z=\mathbb{CP}^{n-1}$. This is equivalent to the claim that, for regular Reeb vector fields, the Bishop inequality never obstructs the existence of a compatible Sasaki–Einstein structure. The source presents it as a conjecture; its resolution is not indicated here.