Euclidean distortion conjecture for special linear groups over finite fields

Let $\mathsf{SL}_k(\mathbb{F}_\mathfrak{q})$ carry its word metric, where $\mathfrak{q}$ is a prime power, and let $\mathsf{c}_{\ell_2}$ denote the least bi-Lipschitz distortion of an embedding into a Hilbert space $\ell_2$. The notation $A\asymp_\mathfrak{q}B$ means that the two quantities are comparable up to positive constants depending only on $\mathfrak{q}$.

Special linear group Euclidean distortion conjecture. For every $k\in\mathbb{N}$ and every prime power $\mathfrak{q}$,

$$\mathsf{c}_{\ell_2}\big(\mathsf{SL}_k(\mathbb{F}_\mathfrak{q})\big)\asymp_\mathfrak{q}\frac{k^{3/2}}{\log k}.$$

The lower-bound order is established for average distortion in the surrounding discussion, and the conjecture asserts that the same asymptotic behavior holds for bi-Lipschitz embeddings. The source suggests a representation-theoretic approach but gives no resolution.