Embedded Calabi–Yau conjecture for finite genus

Let $M\subset\mathbb{R}^3$ be a connected, complete embedded minimal surface of finite genus with compact, possibly empty, boundary. Embedded Calabi–Yau conjecture for finite genus. The surface $M$ is properly embedded in $\mathbb{R}^3$. The source describes this as a fundamental conjecture and as a strong converse to the existence conjecture for open surfaces with compact boundary; no resolution is stated.