of the two inequalities above, if and only if $\Omega$ is Zoll.

If the real Euler class of the bundle associated with $\Omega_*$ vanishes, then the inequality in the conjecture is equivalent to $$\mathcal A_{\min}(\Omega)\leq0\leq \mathcal A_{\max}(\Omega),\qquad \forall\,\Omega\in \mathcal U,$$ with any of the equalities holding if and only if $\Omega$ is Zoll.

The volume, the action and the Zoll polynomial depend on the choice of reference pair $(\mathfrak p_0,c_0)\in\mathfrak Z_C^0(\Sigma)$. However, thanks to Theorem \[t:oszoll\] we will observe in Remark \[r:indepconj\] that the functional $$\big(P\circ \mathcal A_\Omega-{{\mathfrak{Vol}}}(\Omega)\big):\Lambda(\mathfrak p_{\Omega_*})\to{{\mathbb R}}$$ is independent of such a choice. Hence, Conjecture \[con:lsios\] remains the same when we use another reference pair in $\mathfrak Z_C^0(\Sigma)$.

As mentioned at the beginning of this subsection, Conjecture \[con:lsios\] recovers the contact and symplectic systolic-diastolic inequality. In the [*contact*]{} case, we have $\Omega_*={{\mathrm d}}\alpha_*$ for a contact form $\alpha_*$ so that $C=0$. Up to multiplication by a positive constant, we can assume that $T(\alpha_*)=1$. We take $\mathfrak o_\Sigma=\mathfrak o_{\alpha_*}$ and $c_0=0$. There holds $$P(A)=\frac{1}{n+1}\langle e_0^{n},[M_0]\rangle A^{n+1},$$ where $\langle e_0^{n},[M_0]\rangle=\langle [{\mathrm{\omega}}_*^n],[M_*]\rangle>0$. When $n=1$, we have $\langle e_0,[M_0]\rangle=|H_1^\mathrm{tor}(\Sigma;{{\mathbb Z}})|$. If $\Omega={{\mathrm d}}\alpha$ is $C^{0}$-close to $\Omega_*$, we can take $\alpha$ to be a contact form. We have $${{\mathfrak{Vol}}}({{\mathrm d}}\alpha)=\frac1n\mathrm{Volume}(\alpha),\qquad \mathcal 