Systolic inequality for S^1-invariant contact forms in the case e = 0

Ascertain whether a uniform systolic inequality holds for S^1-invariant contact forms on Seifert bundles with Euler number e = 0; specifically, determine whether there exists a constant C > 0 such that for every invariant contact form α on any Seifert bundle with e = 0, sys(α)^2 ≤ C·Vol(α).

Background

The theorem proved in the paper requires non-zero Euler number to obtain a uniform systolic inequality for invariant contact forms. The zero Euler number case—bundles finitely covered by trivial circle bundles—is not addressed by the main result and is stated by the authors to be unresolved.

Resolving this would clarify whether symmetry via an almost-free S^1-action suffices to enforce uniform systolic control even when the Euler number vanishes.