We construct metrics Xn of arbitrarily small volume on the product S1×S2 of the circle and the 2-sphere, verifying the inequality length(γ)·area(Σ)≥1, for every noncontractible curve γ and every homologically nontrivial surface Σ in S1×S2 (a product metric cannot have such a property). In other words, the intersystolic inequality for S1×S2 is false (in contrast with the stable intersystolic inequality). We construct the metrics Xn, which violate the inequality, by surgery on a left-invariant metric on the Heisenberg group. The Xn are Riemannian submersions of cohomogeneity +1, whose fibers circle n times a closed loop in the moduli space of flat tori. The Xn are inspired by M. Gromov's homogeneous metrics on S1×S3 violating the intersystolic inequality. The Xn have an alternative description as "twisted" warped product metrics, where the S2-factor is deformed following a certain flow, giving rise to a non-conformal deformation. We obtain a lower bound for the 2-systole of Xn either by calibration by a truncated area form, or by the use of an area-decreasing projection of the universal cover ∼Xn. The lower bound for the 1-systole relies on a suitable map to a 3-dimensional compact cell complex containing the standard nilmanifold of the Heisenberg group.