Average embedding conjecture for finite metric spaces into iterated $ll_p$ spaces

Let $(\mathcal{M},d_\mathcal{M})$ be an $n$-point metric space. For $p,D\geqslant 1$, a bi-Lipschitz distortion-$D$ embedding of $\mathcal{M}$ into a metric space means an embedding whose Lipschitz distortion is at most $D$.

There is a universal constant $\mathsf{C}\geqslant 1$ such that, for every $p,D\geqslant 1$, there exists $n_0=n_0(p,D)\in\mathbb{N}$ with the following property: whenever $n\geqslant n_0$, every $n$-point metric space $(\mathcal{M},d_\mathcal{M})$ admits a bi-Lipschitz distortion-$D$ embedding into $\ell_p(X)$ for some $k$-dimensional normed space $(X,\|\cdot\|_X)$, where $k\leqslant n^{\mathsf{C}/(pD)}$. Conceivably, this holds even with $X=\ell_\infty^k$.

This conjecture predicts sharp dependence on $p$ in the preceding average John theorem and could have algorithmic applications; the stronger assertion with $X=\ell_\infty^k$ is also left open.