Meeks–Sullivan parabolicity conjecture for finite-genus proper minimal surfaces

Let $f:M\rightarrow\mathbb{R}^3$ be a complete proper minimal immersion, where $M$ is a Riemannian surface without boundary and with finite genus. A Riemannian surface is parabolic if it is noncompact and carries no negative non-constant subharmonic function.

Meeks–Sullivan conjecture. The surface $M$ is parabolic.

This conjecture concerns conformal restrictions on complete proper minimal immersions in $\mathbb{R}^3$. The surrounding results show parabolicity under additional hypotheses, but the source provides no evidence that this finite-genus case has been resolved.