Does there exist a map between the (half-open) square and the (half-open) circle which preserves path lengths?
Such a map would be called an equilong map. Specifically I am searching for a map $f:[0,1)^2 \to D$ (where $D$ is the half-open unit disk defined below) such that for any continuous path $\gamma :[0,1]\to [0,1)^2$,
$$m\{\gamma ([0,1])\}=m\{(f\circ \gamma )([0,1])\} \text{ where $m$ is the Lebesgue measure.}$$
I do NOT want to assume $f$ is differentiable almost everywhere, though if that were true this would mean
$$\int_\gamma dt = \int_{f \circ \gamma}dt, $$ i.e. $$\int_0^1 |\gamma '(t)|dt = \int_0^1 |f'(\gamma (t)) \gamma '(t)|dt.$$
$D$ here is defined as the open unit disk union with its upper-boundary:
$$D=\{(x,y)\in \mathbb{R}^2:x^2+y^2<1\}\cup \{(x,y)\in \mathbb{R}^2:x^2+y^2=1 \text{ and } x>0\}$$
Note that I am NOT requiring that $f$ be continuous! (Though it might be nice for the discontinuities to have measure zero).
I have found sources with several examples of maps which preserve shape vaguely, but none which are equilong, to my knowledge. Here is a link to an example of a map which preserves (fractional) area but does not preserve lengths: https://pdfs.semanticscholar.org/4322/6a3916a85025acbb3a58c17f6dc0756b35ac.pdf.
I have tried to construct one myself but don't have a clue where to begin. I suspect one exists, but have no proof - a non-constructive proof of existence is also fine!