Halperin has conjectured that the Serre spectral sequence of any fibration that has fibre space a certain kind of elliptic space should collapse at the $E_2$-term. In this paper we obtain an equivalent phrasing of this conjecture, in terms of formality relations between base and total spaces in such a fibration (Theorem 3.4). Also, we obtain results on relations between various numerical invariants of the base, total and fibre spaces in these fibrations. Some of our results give weak versions of Halperin’s...