The $C^1$ unipotent Jacobian univalence conjecture

Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a $C^1$ map, and let $J(f)$ denote its Jacobian matrix. A matrix is unipotent when all its eigenvalues are $1$. The $C^1$ unipotent Jacobian univalence conjecture. If $J(f)$ is unipotent, then $f$ is injective. The conjecture is known for $n=2$ and, if established for all $n$, would imply the Jacobian Conjecture. The paper notes that the stronger bijectivity version has no known counterexample.