Let $d\geq 2$. A map $f:\mathbb{R}^d\to\mathbb{R}^d$ is called conformal at a point $x\in\mathbb{R}^d$ if $Df(x)=\lambda U$, where $\lambda>0$ and $U$ is an orthogonal map (that definition includes orientation reversing maps which to exclude we have to assume that $U$ has positive determinant). Holomorphic maps in the complex plane are examples of maps that are conformal everywhere (except at their critical points). In higher dimensions now [Liouville's theorem](https://en.wikipedia.org/wiki/Liouville%27s_theorem_(conformal_mappings)) says that a map $f:G\to\mathbb{R}^d$ that is conformal in a domain $G\subset \mathbb{R}^d$ is equal with a Mobius transformation on that domain.

My question is: Are there any examples of maps defined in some domain in $\mathbb{R}^d$, $d> 2$ that are conformal Lebesgue almost everywhere in the domain of definition but are not Mobius transformations? Is that even possible?