Finiteness conjecture for constant mean curvature surfaces in flat 3-tori

Let $H>0$ and $g\in\mathbb{N}\cup\{0\}$. Consider the moduli space of non-congruent, connected, closed $H$-surfaces of genus at most $g$ in a fixed flat $3$-torus. Finiteness conjecture. This moduli space is finite.

The conjecture is motivated by compactness for bounded-genus constant mean curvature surfaces. It contrasts with the existence of embedded connected closed minimal surfaces of arbitrarily large area in many genera, and the finiteness of the positive-mean-curvature moduli space remains open.