The rank conjecture for injective map germs from surfaces to three-space

Let $g: (\mathbb{C}^2,0)\to(\mathbb{C}^3,0)$ be an injective map germ, and let $\operatorname{grad} g$ denote its gradient. The rank conjecture. The rank of the gradient at the origin satisfies

$$\operatorname{rank}(\operatorname{grad} g)(0)\geq 1.$$

This is presented as an equivalent formulation of Lê Dũng Tráng's real-link conjecture. The source says that there is no proof or disproof, and notes that the claim is delicate even in particular cases.