Halperin conjecture

In rational homotopy theory, the Halperin conjecture concerns the Serre spectral sequence of certain fibrations. It is named after the Canadian mathematician Stephen Halperin.

Statement
Suppose that $$ F \to E \to B $$ is a fibration of simply connected spaces such that $$ F $$ is rationally elliptic and $$ \chi(F) \neq 0 $$ (i.e., $$ F $$ has non-zero Euler characteristic), then the Serre spectral sequence associated to the fibration collapses at the $$ E_2 $$ page.

Status
As of 2019, Halperin's conjecture is still open. Gregory Lupton has reformulated the conjecture in terms of formality relations.