Euclidean distortion conjecture for finite subsets of $L_p$

For $n\to\finity$ and $p>2$, let $\mathsf{c}_2^n(L_p)$ denote the maximum, over all $n$-point metric spaces, of the least distortion with which the space embeds into $L_p$ and then into Euclidean space. The known bounds give $\mathsf{c}_2^n(L_p)=o(\log n)$ when $p=o\big(\sqrt[6]{\log n}/\sqrt[3]{\log\log n}\big)$, while $\mathsf{c}_2^n(L_p)\asymp\log n$ when $p\gtrsim\log n$. Euclidean distortion conjecture. One should have

$$\mathsf{c}_2^n(L_p)=o(\log n)$$

in the remaining range

$$\sqrt[6]{\log n}/\sqrt[3]{\log\log n}\lesssim p=o(\log n).$$

This would determine the qualitative Euclidean distortion growth throughout the currently unresolved intermediate range; the endpoint regimes are supplied by the theorem and by the embedding of $\ell_\infty^n$ into $L_p$.