Halperin's conjecture on the collapse of the rational Serre spectral sequence

Let $X$ be an $F_0$-space, meaning an elliptic space with evenly graded rational cohomology, and let $p \colon E \to B$ be any fibration of simply connected spaces with fibre $X$. Halperin's conjecture. The rational Serre spectral sequence for $p$ collapses at the $E_2$-term. Halperin's conjecture generalizes classical results on the rational cohomology of homogeneous spaces. It is used here to extend Stanley's calculation of the rational sectional category of the universal fibration with fibre an even-dimensional sphere to $F_0$-spaces satisfying the conjecture; its general status is not established by the supplied text.