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. 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]

regularif 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 $s$-regularfor 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 $X$.