GMSY's Fano intersection-number conjecture

Let $V$ be a smooth Fano manifold of complex dimension $n-1$, and let $I(V)\in\mathbb{N}$ be its Fano index. GMSY's Fano inequality conjecture.

$$I(V)\int_V c_1(V)^{n-1}\leq n\int_{\mathbb{CP}^{n-1}}c_1(\mathbb{CP}^{n-1})^{n-1}=n^n,$$

with equality if and only if $V=\mathbb{CP}^{n-1}$.

The source states this as the smooth Fano reformulation of the regular Reeb-field volume conjecture. No resolution is supplied in the paper.