Martio's conjecture on branching of low-distortion quasiregular maps

Let $\Omega\subset\mathbb R^n$ be a domain, let $f\colon\Omega\to\mathbb R^n$ be a non-constant quasiregular map, and write $B_f$ for its branch set, the set of points where $f$ is not a local homeomorphism. Define the inner distortion by

$$K_I(f)=\sup_{x\in\Omega}K_I(f,x),$$

where

$$K_I(f,x)= \begin{cases} \dfrac{|\operatorname{cof}(\mathrm{D}f(x))|^n}{(\det\mathrm{D}f(x))^{n-1}}&\text{if }\det\mathrm{D}f(x)>0,\\ 1&\text{otherwise}. \end{cases}$$

Martio's conjecture. If $n\geq3$ and $K_I(f)<2$, then

$$B_f=\varnothing.$$

The conjecture is a stability form of Liouville's theorem, asserting that sufficiently small inner distortion prevents branching in dimensions at least three. It remains open; Rajala proved the weaker general result that $K_I(f)<1+\varepsilon(n)$ implies $B_f=\varnothing$.