A_{{{\mathrm d}}\alpha}(\gamma)=\int_{S^1}\gamma^*\alpha=T(\gamma),\quad \forall\,\gamma\in\Lambda_{\mathfrak h}(\Sigma).$$ Thus, there holds $T_{\min}(\alpha,\mathfrak h)\leq \mathcal A_{\min}({{\mathrm d}}\alpha)\leq \mathcal A_{\max}({{\mathrm d}}\alpha)\leq T_{\max}(\alpha,\mathfrak h)$, and Conjecture \[con:lsios\] implies $$T_{\min}(\alpha,\mathfrak h)^n\leq \frac{1}{\langle e_0^{n},[M_0]\rangle}\mathrm{Volume}(\alpha)\leq T_{\max}(\alpha,\mathfrak h)^n$$ with equality cases exactly when the contact form $\alpha$ is Zoll.

In the [*symplectic*]{} case, $\mathfrak p_{\Omega_*}$ is trivial so that $\Sigma=M_*\times S^1$ and we call $t\in S^1$ the global angular coordinate. We take $\Omega=\Omega_*+{{\mathrm d}}(H{{\mathrm d}}t)$ for some function $H:M_*\times S^1\to {{\mathbb R}}$. The characteristic distribution of $\Omega$ is generated by the vector field $\partial_t+X_{H_t}$, where $X_{H_t}$ is the $\omega_*$-Hamiltonian vector field of $H_t:=H(\cdot,t)$ tangent to $M_*\times\{t\}$. Then, $\mathcal A_\Omega$ is the Hamiltonian action defined in , $\langle c_0^n,[M_0]\rangle=\langle [{\mathrm{\omega}}_*^n],[M_*]\rangle>0$, and $$P(A)=\langle c_0^n,[M_0]\rangle A,\qquad {\mathrm{Vol}}(H{{\mathrm d}}t)=\int_{M_*\times S^1} (H{{\mathrm d}}t)\wedge\omega_*^{n}={\mathrm{CAL}}_{{\mathrm{\omega}}_*}(H).$$ Conjecture \[con:lsios\] generalises Proposition \[p:hamsys\], since $\mathcal A_{\min}(\Omega)=\min\mathcal A_H$ and $\mathcal A_{\max}(\Omega)=\max\mathcal A_H$, if we take the Hamiltonian $H$ to be normalised, namely ${\mathrm{CAL}}_{{\mathrm{\omega}}_*}(H)=0$.

Inspired by the Hamiltonian case, we will look first at a special class of $\Omega\in\Xi^2_C(\Sigma)$ when studying Conjecture \[con:lsios\]. Here, we assume without loss of generality that $\mathfrak p_0=\mathfrak p_{\Omega_*}$. We fix an $S^1$-connection form $\eta$ for the bundle $\mathfrak p_0$. This means that $\eta$ restricts to the angular form on each fibre and ${{\mathrm d}}\eta=\mathfrak p_0^*\kappa$ for some 