Cubical area growth conjecture for finite-genus constant-mean-curvature surfaces

Let $M$ be a complete surface of finite genus embedded in $\mathbb{R}^3$ with constant mean curvature. Say that $M$ has at most cubical area growth if there is a constant $C$ depending on $M$ such that, for every ambient ball of radius $R$, the area of $M$ inside that ball is less than $CR^3$. Cubical area growth conjecture. Such an $M$ has at most cubical area growth. In particular, every such surface is properly embedded in $\mathbb{R}^3$. This conjecture is motivated by the preceding properness result for complete embedded constant-mean-curvature surfaces of finite topology; the supplied source does not state whether the finite-genus claim has been resolved.