Existence of invariant contact forms with unbounded systolic ratio as Euler number tends to zero

Determine whether there exists a sequence of Seifert bundles M_n with Euler numbers e_n → 0, each admitting an S^1-invariant contact form α_n, such that the systolic ratios ρ(α_n) = sys(α_n)^2 / Vol(α_n) diverge to +∞.

Background

The main result establishes a systolic inequality for S^1-invariant contact forms on Seifert bundles with non-zero Euler number e: sys(α)^2 ≤ C·max(1,1/|e|)·Vol(α). As |e|→0, the bound on the right-hand side grows, suggesting the possibility of contact forms with increasingly large systolic ratio.

The authors explicitly highlight uncertainty about whether such growth is realized in practice by invariant contact forms on Seifert bundles as their Euler number tends to zero.