Low-dimensional embedding conjecture for special linear groups

Fix a prime power $\mathfrak{q}$. For $D\geqslant1$, let $\operatorname{dim}_D(k,\mathfrak{q})$ be the least dimension of a normed space into which the word-metric group $\mathsf{SL}_k(\mathbb{F}_\mathfrak{q})$ embeds with bi-Lipschitz distortion $D$.

Low-dimensional special linear group conjecture. For every prime power $\mathfrak{q}$ there exist $D=D(\mathfrak{q})\geqslant1$ and positive constants $c=c(\mathfrak{q})$, $C=C(\mathfrak{q})$ such that, for every integer $k\geqslant2$,

$$e^{ck/\log k}\leqslant \operatorname{dim}_D(k,\mathfrak{q})\leqslant e^{Ck/\log k}.$$

The lower bound is proved in the surrounding corollary, while the conjecture asks for a matching constant-distortion upper bound. Such an embedding would improve substantially on the exponential-in-$k^2$ dimension supplied by Fréchet's embedding.