Properness and cubic area growth conjecture for finite-genus constant mean curvature surfaces

Let $N$ be a complete flat $3$-manifold, and let $M$ be a complete connected $H$-surface embedded in $N$ with finite genus. Let $W$ denote its closed mean-convex complement when $M$ is proper, and let $I\in(0,\infty]$ be a lower bound for the injectivity radius of $N$. Properness and cubic area growth conjecture. The following hold:

1. $M$ is proper in $N$; consequently, $M$ separates $N$ and $W$ is a handlebody of radius at most $1/H$.
2. There exists $C_{H,I}>2$, depending only on $H$ and $I$, such that for $R\geq 2/H$ and $p\in W$,

$$\frac{1}{C_{H,I}}\operatorname{Area}(M\cap B_N(p,R))\leq \operatorname{Volume}(W\cap B_N(p,R))\leq \operatorname{Volume}(B_N(p,R))\leq \frac{4}{3}\pi R^3.$$

In particular, when $N=\mathbb{R}^3$, $M$ has at most cubical area growth: its area in ambient balls of radius $R>2/H$ is at most $\frac{1}{H}C_{1,\infty}R^3$. This is an outstanding problem concerning properness and quantitative area growth for finite-genus constant mean curvature surfaces in flat $3$-manifolds.