Gałęski's local smooth rank-constrained approximation conjecture

Let $1\leq m<n$ be integers and let $\Omega\subset\mathbb R^n$ be open. If $f\in C^1(\Omega,\mathbb R^n)$ satisfies $\operatorname{rank} Df\leq m$ everywhere in $\Omega$, then for every point $x\in\Omega$ there is a neighborhood $\mathbb B^n(x,\varepsilon)\subset\Omega$ and a sequence $f_i\in C^\infty(\mathbb B^n(x,\varepsilon),\mathbb R^n)$ such that $\operatorname{rank} Df_i\leq m$ and $f_i$ converges to $f$ uniformly on $\mathbb B^n(x,\varepsilon)$.

Gałęski's local approximation conjecture. Every such mapping admits, around each point, a uniformly convergent smooth approximation sequence preserving the derivative-rank bound.

The paper states that this weaker local form is also false in general, with the main result providing counterexamples for certain ranges of $n$ and $m$.