Local systolic-diastolic inequality for odd-symplectic forms

Let $\Sigma$ be the underlying manifold, let $C\in H^2_{\mathrm{dR}}(\Sigma)$, and let $\Omega_*$ be a Zoll odd-symplectic form with cohomology class $C$. For forms $\Omega\in\mathcal S_C(\Sigma)$ sufficiently close to $\Omega_*$, let $\mathcal A_{\min}(\Omega)$ and $\mathcal A_{\max}(\Omega)$ denote the minimum and maximum of the action over the relevant periodic characteristics, and let $P$ be the polynomial associated with $\Omega_*$. Local systolic-diastolic inequality. There is a $C^{k-1}$-neighbourhood $\mathcal U$ of $\Omega_*$ in $\mathcal S_C(\Sigma)$, with $k\geq 2$, such that

$$P(\mathcal A_{\min}(\Omega))\leq{\mathfrak{Vol}}(\Omega)\leq P(\mathcal A_{\max}(\Omega)),\qquad \forall\,\Omega\in\mathcal U.$$

Equality in either inequality holds if and only if $\Omega$ is Zoll. This is a local systolic-diastolic inequality near a Zoll odd-symplectic form; the supplied text does not state whether the conjecture has been proved or disproved.