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

Let $X$ be a toric Fano $d$-fold with $d\geq 3$. Call $X$ non-splittable if it is not a product of two lower-dimensional toric Fano manifolds, and let $\rho_X$ denote its Picard number.

Non-splittable extremal conjecture. One has

$$\rho_X\leq \frac{4d+3}{3},$$

with equality if and only if $d=3d'$ for a positive integer $d'$ and $X$ is isomorphic to a uniquely determined toric $(S_3)^{2d'}$-bundle over $\mathbb P^{d'}$.

This is proposed as a refinement of Casagrande's upper bound, motivated by classification results in low dimensions. The source gives no resolution, so the conjecture remains open.