I want to ask about the progress on **Question 8** from ["Thirty-three yes or no questions about mappings, measures, and metrics" by Juha Heinonen and Stephen Semmes][1]. Is it still open? If yes, was some progress achieved? Is it known to be especially hard?

Here is the question:

> **Question 8**. If an Ahlfors regular metric space admits a [David–Semmes] regular map into some
Euclidean space, then does it admit a bi-Lipschitz map into another, possibly different, Euclidean space?

And definitions:

> A map from one metric space $X$ into another metric space $X'$ is said to be [David–Semmes] **regular** if it is Lipschitz and if there is a constant $C \ge 1$ so that the preimage of each ball of radius $R$ in $X'$ can be covered by at most $C$ balls of radius $R$ in $X$.

> A metric space $X$ is said to be **Ahlfors $s$-regular** for some real number $s > 0$ if
it has Hausdorff dimension $s$ and if there is a constant $C \ge 1$ such that
$$C^{−1}R^s ≤ H_s(B_R) ≤ CR^s$$
for each metric ball $B_R$ of radius $R < diam X$. Here $H_s$ denotes the $s$-dimensional
Hausdorff measure in $X$. 


  [1]: http://www.ams.org/journals/ecgd/1997-01-01/S1088-4173-97-00012-X/S1088-4173-97-00012-X.pdf