Suppose $f : \mathbb{R}^n \to \mathbb{R}^n$ has Jacobian $Jf : \mathbb{R}^n\to\mathsf{M}_n(\mathbb{R})$ with positive eigenvalues everywhere. Is $f$ (globally) injective (invertible on its range)?
If $Jf$ was also guaranteed to be symmetric, this would be true by this question. We also know $f$ is locally invertible by the inverse function theorem.