The two-fixed-point Jacobian eigenvalue conjecture

Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a $C^1$ map, and let $J(f)$ denote its Jacobian matrix. A point $x$ is a fixed point if $f(x)=x$. The two-fixed-point conjecture. If $f$ has two distinct fixed points, then $J(f)$ has a nonzero eigenvalue at some point. The paper states that this is equivalent to the $C^1$ unipotent Jacobian univalence conjecture; it is therefore open in the generality considered.