The linear Picard-number bound for toric Fano varieties

Let $X$ be an $n$-dimensional toric Fano variety, and let $\rho$ denote its Picard number. The preceding discussion suggests that varieties with maximal Picard number are $S_3$-bundles over toric Fano varieties of lower dimension, so that induction applies. Linear Picard-number conjecture. One should have

$$\rho\leq 2n \text{ for } n \text{ even},\qquad \rho\leq 2n-1 \text{ for } n \text{ odd}.$$

This would give a linear bound in the dimension for the Picard number of toric Fano varieties, improving the previously known general bounds. The source presents it conditionally on the asserted bundle structure and does not provide a resolution.