The image of totally disconnected compacta under Lipschitz maps

Question

The Alexandroff-Hausdorff theorem tells that every compact metric space is the continuous image of the Cantor set. I am wondering whether the image of a compact totally disconnected subset in $\mathbb R^n$ under a Holder-Lipschitz map $L:\mathbb R^n\to \mathbb R^m$ is totally disconnected again. I know only that sets of measure zero go to sets of measure zero for $m=n$, but this seems to be unrelated.

Answer 1

This answer gives an example of a Lipschitz map with constant $1$ such that the domain is a Cantor set (so totally disconnected) and the image a space homeomorphic to $[0,1]$. So it seems no?