Conjecture on realizing arbitrary-genus symplectic leaves

Let $k\in\mathbb{N}$. Consider the family of Poisson brackets described by, with $U$ chosen appropriately and $V=0$, and let the resulting symplectic leaves be the corresponding surfaces.

Arbitrary-genus symplectic-leaf conjecture. For every integer $k\in\mathbb{N}$, an appropriate choice of $U$ produces a sample of symplectic leaves that are topologically Riemann surfaces of genus $k$, with optionally one or two boundary components.

This conjecture proposes that the freedom in the function $U$ allows the topology of symplectic leaves to realize arbitrary genus, extending the examples with one- and two-punctured surfaces described above. The supplied text gives no evidence of resolution.