3 problems

Gałęski's conjecture. If $f\in C^1(\Omega,\mathbb R^n)$ satisfies

Rank-one approximation conjecture. If $m=1$, then uniform approximation is possible by smooth mappings whose derivative has rank at most $1$.

Local rank-constrained approximation conjecture. If $f\in C^1(\Omega,\mathbb R^n)$ satisfies