Polynomial-dimensional embedding conjecture for finite subsets of $\ell_1$

Let $\mathcal{M}\subseteq \ell_1$ be a subset with $|\mathcal{M}|=n$. An embedding of $\mathcal{M}$ into a normed space has distortion $D$ if distances are preserved up to a multiplicative factor with ratio $D$. Embedding conjecture. Every $n$-point subset of $\ell_1$ embeds with distortion $O(1)$ into some normed space of dimension $(\log n)^{O(1)}$. This would improve the known lower-bound framework for dimensionality reduction into general normed spaces and would show that the lower bound of Matoušek cannot occur for subsets of $\ell_1$; the conjecture is presented as open.