Sharp distortion conjecture for $L_q$ integer grids in $L_p$

For $2<q<p$ and $m,n\in\mathbb N$, let $[m]_q^n$ denote the $n$-dimensional integer grid with its $\ell_q$ metric, and let $c_p([m]_q^n)$ be its least bi-Lipschitz distortion in $L_p$. Let $J^R_{(q\to p;n)}$ and $J^S_{(q\to p;n)}$ be the Rosenthal and Schoenberg embeddings introduced in the source. Sharp grid distortion conjecture. The better of these two embeddings is optimal up to constants depending only on $p$ and $q$; equivalently,

$$c_p([m]_q^n)\asymp_{p,q}\min\left\{n^{\frac{(p-q)(q-2)}{q^2(p-2)}},m^{1-\frac{2}{q}}\right\}.$$

In particular, there is $\eta(p,q)\in(0,1)$ such that $m\geqslant n^{\frac{p-q}{q(p-2)}}$ implies $c_p([m]_q^n)\geqslant\eta(p,q)c_p(\ell_q^n)$, whereas $m=o(n^{\frac{p-q}{q(p-2)}})$ implies $c_p([m]_q^n)=o(c_p(\ell_q^n))$. The conjecture compares the two known constructions and is explicitly left open.