Rank-one smooth approximation conjecture

Let $n,k\geq 2$ and let $f\in C^1(\mathbb R^n,\mathbb R^k)$ satisfy $\operatorname{rank} Df\leq 1$.

Rank-one approximation conjecture. The mapping $f$ can be uniformly approximated, at least locally, by mappings $g\in C^\infty(\mathbb R^n,\mathbb R^k)$ satisfying $\operatorname{rank} Dg\leq 1$.

The paper presents this as a special case believed to be true after disproving the general conjecture. Its resolution is not supplied in the source.