Hock's x-y swap conjecture for rational spectral curves

Let $\Sigma=\mathbb{C}\mathrm{P}^1$. In Case 1, let $y(z)=z$ be a global affine coordinate and let $x=\log R(z)+Az$, where $R(z)$ is a rational function and $A\in\mathbb{C}$. In Case 2, let $y(z)=\log z$, where $z$ is a global affine coordinate, and let $x=\log R(z)$, with $R(z)$ again rational. Assume that all zeros of $dx$ are simple. Define the dual system formally by

$$\sum_{g=0}^\infty \hbar^{2g}\omega^{\vee,(g)}_{1}=-dy(z)\frac{1}{\mathcal{S}(\hbar\partial_{y(z)})}x(z),$$

and

$$\omega^{\vee,(g)}_n=0$$

for $g\geq 0$, $n\geq 2$, and $2g-2+n>0$. Hock's x-y swap conjecture. The correlation differentials $\{\omega^{(g)}_n\}$ can be obtained by the $x-y$ swap relation from this dual system of differentials. This conjecture proposes a general x-y swap description for these rational spectral curves; its resolution is not specified in the source.