The Lang-Plaut problem: does every doubling subset of Hilbert space bi-Lipschitz embed in some Euclidean space?

Claims a negative answer: a fixed doubling subset of $\ell_2$ (doubling constant at most 76800) with no bi-Lipschitz embedding into any $\mathbb R^k$; more generally every infinite-dimensional real Banach space contains a compact set of universally bounded doubling constant with no bi-Lipschitz embedding into any finite-dimensional normed space, and the same metric embeds isometrically in every $L_p$, $1\le p<\infty$. It does NOT bear on snowflaked embeddings (Assouad). Priority note: Schioppa announced a negative answer in 2017 (arXiv:1703.10265) but withdrew it.

Claims a negative answer: a fixed doubling subset of $\ell_2$ (doubling constant at most 76800) with no bi-Lipschitz embedding into any $\mathbb R^k$; more generally every infinite-dimensional real Banach space contains a compact set of universally bounded doubling constant with no bi-Lipschitz embedding into any finite-dimensional normed space, and the same metric embeds isometrically in every $L_p$, $1\le p<\infty$. It does NOT bear on snowflaked embeddings (Assouad). Priority note: Schioppa announced a negative answer in 2017 (arXiv:1703.10265) but withdrew it.