The one-two-three conjecture for genus-three minimal surfaces in flat three-tori

Let $M$ be a compact oriented minimal surface in a flat three-torus, and suppose that $M$ has genus three. Write $index_A$ for its Morse index with respect to the area functional. One-two-three conjecture. One has

$$1\leq index_A\leq 3.$$

The preceding results establish this bound for the five described families and indicate that genus-three compact oriented minimal surfaces in flat three-tori should have uniformly bounded Morse index. Whether the bound holds for every such surface remains open.