Halperin's conjecture on fibrations of F₀-spaces

Let an $F_0$-space be a finite simply connected complex with finite-dimensional rational homotopy, such that $H^{\mathrm{odd}}(X,\mathbb{Q})=0$. Let $X\to E\to B$ be a fibration of simply connected spaces, with fibre inclusion $j\colon X\to E$.

Halperin's conjecture. The fibration is TNCZ, meaning that $j$ induces a surjection on rational cohomology.

This is a well-known conjecture in rational homotopy theory concerning the behaviour of fibrations with $F_0$-space fibres. The source presents it as an open conjecture and uses it to relate the rationalized $G$-sequence to rational cohomology.