Yamaguchi–Yokura's estimate for the homotopical-to-cohomological quotient in elliptic fibrations

Let $F\hookrightarrow X\to B$ be a fibration of rationally elliptic spaces, and let $h$ denote the quotient studied by Yamaguchi and Yokura. The product fibration has total space $F\times B$.

Yamaguchi–Yokura's conjecture.

$$\frac{1}{2}\cdot h(F\times B) \leq h(X) < h(F)+h(B)+\frac{1}{4}$$

This conjecture concerns estimates for the quotient $h$ in fibrations of rationally elliptic spaces. The paper proves several special cases and asymptotic results, but the general estimate is not resolved here.