Hoffman–Meeks finite topology conjecture for properly minimally embedded surfaces

Let $M$ be a connected surface of finite topology, with genus $g$ and $r>2$ ends. Finite topology conjecture. The surface $M$ can be properly minimally embedded in $\mathbb{R}^3$ if and only if

$$r\leq g+2.$$

By Collin's theorem, a properly embedded minimal surface of finite topology with finitely many ends has finite total curvature. The conjecture is motivated by the known abundance of such examples; even in the simplest case $g=1$, it remains open.