Snowflake embedding conjecture for finite-dimensional normed spaces

For $\upomega\in(0,1]$, the $\upomega$-snowflake of a metric space is obtained by raising its distance to the power $\upomega$. A quadratic average distortion embedding into $\ell_2$ is an embedding whose distortion is measured using the quadratic, or $p=2$, average.

Finite-dimensional snowflake embedding conjecture. For every $\upomega\in(0,\tfrac12)$, there exists $\mathsf{C}_\upomega>0$ such that, for every integer $k\geqslant 2$, the $\upomega$-snowflake of every $k$-dimensional normed space embeds into $\ell_2$ with quadratic average distortion at most $\mathsf{C}_\upomega(\log k)^\upomega$.

The conjecture would improve the currently obtained $O_\upomega(\sqrt{\log k})$ bound and, according to the source, would yield an asymptotically sharp estimate for fixed $\upomega$ as $k\to\infty$.