Ergodicity conjecture for anti-symmetric skew products over locally Hamiltonian flows

Let $\psi_\mathbb{R}$ be a locally Hamiltonian flow on a compact surface $M$ for which Theorem~ holds. Let $M'$ be one of its minimal components of genus $g\geq 1$, and set

$$m:=\max\{m_\sigma:\sigma\in\operatorname{Fix}(\psi_\mathbb{R})\cap M'\}.$$

For $f\in C^m(M)$, suppose that $D_i(f)=0$ for all $1\leq i\leq g$, that $\mathfrak{C}^k_{\sigma,\ell}(f)=0$ for every $\sigma\in\operatorname{Fix}(\psi_\mathbb{R})\cap M'$, $0\leq k<m_\sigma-2$, and $\ell\in\mathcal{L}^\sigma_\sim$, and that

$$\mathfrak{C}^{m_\sigma-2}_{\sigma,\ell}(f)\neq 0$$

for some $\sigma\in\operatorname{Fix}(\psi_\mathbb{R})\cap M'$ and $\ell\in\mathcal{L}^\sigma_\sim$. Working conjecture. Under these conditions, the skew product flow $\psi^f_\mathbb{R}$ is ergodic.

The conjecture proposes that the symmetry condition on logarithmic singularities is unnecessary for ergodicity when the relevant lower-order obstructions vanish and the first nonvanishing singularity coefficient occurs at order $m_\sigma-2$. It would extend known ergodicity results for symmetric logarithmic singularities to minimal components with saddle loops, where symmetry is naturally broken. The source does not state a resolution, so the conjecture is treated as open.