We say, following this definition, that a domain $\Omega\subset \mathbb{R}^{n}$ is *weakly Lipschitz* if it can locally be flattened by a Lipschitz homeomomorphism $\phi$ (i.e., a Lipschitz continuous map $\phi$ such that its inverse $\phi^{-1}$ is still Lipschitz).

My question is the following: given an hyperplane $H\subset \mathbb{R}^n$, we consider the projection $\pi_H:\mathbb{R}^n\to H$. Then, is it true that $\pi(\Omega)$ is still a weakly Lipschitz domain?

I've found here that the answer is negative for *strongly Lipschitz* domains, and I don't have enough intuition to see what can possibly go wrong with weakly Lipschitz domain (although I'm afraid there can be something weird)

Any reference or help is welcomed. Thank you!