In some sense, this is probably the most unsatisfactory answer possible, but it's all I know:

In the 2005 paper 1, Movahedi-Lankarani and Wells write the following in VII (p. 16):

Tomi Laakso (6) has informed us that his non-embeddability theorem [9] can be generalized in such a way that the non-embeddability theorem of Cheeger [4] no longer implies it. In a forthcoming paper, he gives a sufficient condition for a metric space to admit no bi-Lipschitz embedding into uniformly convex Banach spaces. Furthermore, he shows that there exists a metric space which does not admit a bi-Lipschitz embedding into any uniformly convex Banach space even though it admits a David-Semmes regular mapping onto $\mathbb{R}^2$.

Reference [9] is to the paper of Laakso in which he constructs Ahlfors $Q$-regular spaces of arbitrary $Q>1$ admitting Poincare inequalities and not bi-Lipschitz embedding into any uniformly convex Banach space. Unfortunately, the footnote (6) is just ``private communication''. As far as I know, the paper of Laakso mentioned in this quote never came out.

I for one would be happy to see an example like this, but I have not seen one anywhere else. Of course, someone might know better.

1 H. Movahedi-Lankarani and R. Wells, On bi-Lipschitz embeddings Port. Math. (N.S.) 62 (2005), no. 3, 247–268.

Link to MathSciNet, Link to the paper

In some sense, this is probably the most unsatisfactory answer possible, but it's all I know:

In the 2005 paper 1, Movahedi-Lankarani and Wells write the following in VII (p. 16):

Tomi Laakso (6) has informed us that his non-embeddability theorem [9] can be generalized in such a way that the non-embeddability theorem of Cheeger [4] no longer implies it. In a forthcoming paper, he gives a sufficient condition for a metric space to admit no bi-Lipschitz embedding into uniformly convex Banach spaces. Furthermore, he shows that there exists a metric space which does not admit a bi-Lipschitz embedding into any uniformly convex Banach space even though it admits a David-Semmes regular mapping onto $\mathbb{R}^2$.

Reference [9] is to the paper of Laakso in which he constructs Ahlfors $Q$-regular spaces of arbitrary $Q>1$ admitting Poincare inequalities and not bi-Lipschitz embedding into any uniformly convex Banach space (which was published in GAFA). Unfortunately, the footnote (6) is just "private communication". As far as I know, the "forthcoming paper" of Laakso mentioned in this quote never came out.

I for one would be happy to see an example like this, but I have not seen one anywhere else. Of course, someone might know better.

1 H. Movahedi-Lankarani and R. Wells, On bi-Lipschitz embeddings Port. Math. (N.S.) 62 (2005), no. 3, 247–268.

Link to MathSciNet, Link to the paper

In some sense, this is probably the most unsatisfactory answer possible, but it's all I know:

In the 2005 paper 1, Movahedi-Lankarani and Wells write the following in VII (p. 16):

Tomi Laakso (6) has informed us that his non-embeddability theorem [9] can be generalized in such a way that the non-embeddability theorem of Cheeger [4] no longer implies it. In a forthcoming paper, he gives a sufficient condition for a metric space to admit no bi-Lipschitz embedding into uniformly convex Banach spaces. Furthermore, he shows that there exists a metric space which does not admit a bi-Lipschitz embedding into any uniformly convex Banach space even though it admits a David-Semmes regular mapping onto $\mathbb{R}^2$.

Reference [9] is to the paper of Laakso in which he constructs Ahlfors $Q$-regular spaces of arbitrary $Q>1$ admitting Poincare inequalities and not bi-Lipschitz embedding into any uniformly convex Banach space. Unfortunately, the footnote (6) is just ``private communication''. As far as I know, the paper of Laakso mentioned in this quote never came out, and I believe Laakso left mathematics at some point afterwards.

I for one would be happy to see an example like this, but I have not seen one anywhere else. Of course, someone might know better.

1 H. Movahedi-Lankarani and R. Wells, On bi-Lipschitz embeddings Port. Math. (N.S.) 62 (2005), no. 3, 247–268.

Link to MathSciNet, Link to the paper

In some sense, this is probably the most unsatisfactory answer possible, but it's all I know:

In the 2005 paper 1, Movahedi-Lankarani and Wells write the following in VII (p. 16):

Tomi Laakso (6) has informed us that his non-embeddability theorem [9] can be generalized in such a way that the non-embeddability theorem of Cheeger [4] no longer implies it. In a forthcoming paper, he gives a sufficient condition for a metric space to admit no bi-Lipschitz embedding into uniformly convex Banach spaces. Furthermore, he shows that there exists a metric space which does not admit a bi-Lipschitz embedding into any uniformly convex Banach space even though it admits a David-Semmes regular mapping onto $\mathbb{R}^2$.

Reference [9] is to the paper of Laakso in which he constructs Ahlfors $Q$-regular spaces of arbitrary $Q>1$ admitting Poincare inequalities and not bi-Lipschitz embedding into any uniformly convex Banach space. Unfortunately, the footnote (6) is just ``private communication''. As far as I know, the paper of Laakso mentioned in this quote never came out.

I for one would be happy to see an example like this, but I have not seen one anywhere else. Of course, someone might know better.

1 H. Movahedi-Lankarani and R. Wells, On bi-Lipschitz embeddings Port. Math. (N.S.) 62 (2005), no. 3, 247–268.

Link to MathSciNet, Link to the paper