Embeddedness conjecture for the $(m,n,m+n-1)$ family of harmonic ends

Let $m$, $n$ and $k$ be positive integers such that

$$\frac{m-1}{k},\quad \frac{n-1}{k},\quad \frac{m+n-2}{k}$$

are relatively prime, and suppose that $\frac{m-1}{k}$ and $\frac{n-1}{k}$ have different parity. For sufficiently small disks around $0$, define an end by

$$\omega_1=\frac{i}{z^m}\,dz,\qquad \omega_2=\frac{i}{z^n}\,dz,\qquad \omega_3=\frac{1}{z^{m+n-1}}\,dz.$$

Embeddedness conjecture. This end is a complete, properly embedded end of type $(m,n,m+n-1)$. The surrounding discussion says that numerical evidence supports embeddedness, but that no proof is known except in special cases; the conjecture concerns a family that is difficult to incorporate into complete surfaces.