Alexandrov's spectral-radius conjecture for global mappings

Let $n\geqslant 2$ and let $f:\mathbb{R}^n\to\mathbb{R}^n$ be continuously differentiable. Assume that $\det f'(x)\neq 0$ and that the spectral radius of the inverse derivative is uniformly bounded:

$$\rho\bigl(f'(x)^{-1}\bigr)\leqslant A$$

for every $x\in\mathbb{R}^n$, where $A<+\infty$ is constant.

Alexandrov's conjecture. The image $f(\mathbb{R}^n)$ is a convex domain in $\mathbb{R}^n$, and $f$ is injective.

The conjecture extends the planar result discussed in the paper and is presented as unresolved; the weaker injectivity-only version is attributed separately to Chamberland.