Mukai's conjecture for smooth Fano varieties

Let $X$ be a smooth Fano variety of dimension $n$. Its Picard number is $\rho_X\coloneq\operatorname{rk}\operatorname{Pic}(X)$, and its Fano index is

$$i_X\coloneq\max\{k\in\mathbb{Z}_{>0}\mid kH=[-K_X],\ H\in\operatorname{Pic}(X)\}.$$

Mukai's conjecture. One has

$$(i_X-1)\rho_X\le n,$$

with equality if and only if

$$X\cong(\mathbb{P}^{i_X-1})^{\rho_X}.$$

The conjecture gives a numerical characterisation of products of projective spaces among smooth Fano varieties. It remains open in general, although it is proved for several classes, including the class studied in this paper via Cox rings and embeddings into smooth projective toric varieties.