Extremal Picard-number conjecture for non-$S_3$-splittable toric Fano manifolds

Let $X$ be a toric Fano $d$-fold. Call $X$ non-$S_3$-splittable if it is not a product of $S_3$ and a lower-dimensional toric Fano manifold. Let $\rho_X$ denote its Picard number.

Non-$S_3$-splittable extremal conjecture. One has

$$\rho_X\leq \frac{5d}{3},$$

with equality if and only if $d=3d'$ for a positive integer $d'$ and $X$ is isomorphic to the $d'$-fold product of a uniquely determined toric $S_3$-bundle over $\mathbb P^1$.

This refines the product-decomposition conjecture by describing the expected sharp extremal case after all $S_3$ factors have been removed. The source presents it as an open conjecture based on low-dimensional classification results.