Meeks–Pérez–Ros genus-zero conjecture for properly embedded minimal surfaces

Let $M\subset\mathbb{R}^3$ be a properly embedded minimal surface with genus zero and infinitely many ends.

Genus-zero conjecture. $M$ is a Riemann minimal example.

This conjecture extends the characterization of Riemann minimal examples by Meeks, Pérez, and Ros, removing the hypothesis of periodicity. Its status is not established in the supplied source.