this case, we show in Proposition \[p:Moser\] that if $\Omega\in\Xi^2_C(\Sigma)$ is $C^2$-close to $\Omega_*$, then $\Omega$ is an H-form with a $C^2$-small defining Hamiltonian $H$, after applying a diffeomorphism of $\Sigma$ isotopic to the identity. Therefore, in this setting the conjecture follows from Proposition \[p:hamsys\] (or Proposition \[p:qaos\]) and the invariance of the volume and the action under diffeomorphisms. From this result and the topological Lemma \[l:3state1\], Conjecture \[con:lsios\] for bundles with $e_0=0$ can readily be proven.

\[t:triv\] The local systolic-diastolic inequality for odd-symplectic forms, whose associated bundle has vanishing real Euler class, holds true in the $C^2$-topology, .

If $\Omega_*$ is any Zoll odd-symplectic form on a closed three-manifold $\Sigma$, then either the Euler class of its associated bundle vanishes or $\Omega_*={{\mathrm d}}\alpha_*$ for some Zoll contact form $\alpha_*$. Hence, Theorem \[t:triv\] together with [@BK19a Theorem 1.4] establish Conjecture \[con:lsios\] in dimension three.

\[c:mainsys\] The local systolic-diastolic inequality for odd-symplectic forms holds true in the $C^2$-topology on closed three-manifolds.

More generally, one could try to formulate a systolic-diastolic inequality in a neighbourhood of an odd-symplectic form $\Omega_*$, whose closed characteristics are tangent to a (not necessarily free) $S^1$-action on $\Sigma$, cf. [@Tho76] and [@BR94].

### The example of magnetic 