Milnor's conjecture on complete umbilic-free surfaces

Let $S$ be a complete surface immersed in $\mathbb{R}^3$ of class $C^2$ and without umbilic points. Assume that the sum of the squares of its principal curvatures is bounded away from zero.

Milnor's conjecture. Either the Gauss curvature $K$ changes sign on $S$, or $K=0$.

This is described as the most well-known still open problem closely related to Efimov's nonimmersion theorem for complete surfaces of strictly negative curvature.