Sampling-width rate for weighted mixed Sobolev spaces

Let $d\geq2$, let $r\in\mathbb N$, let $\lambda$ be an even integer with $\lambda>4$, and let $\varrho_n(\boldsymbol W^r_2(\mathbb R^d;\mu),L_2(\mathbb R^d;\mu))$ denote the sampling $n$-width of the weighted mixed Sobolev ball in $L_2(\mathbb R^d;\mu)$. Set $r_\lambda$ to be the smoothness parameter associated with $r$ and $\lambda$ in the paper. Sampling-width conjecture.

$$\varrho_n(\boldsymbol W^r_2(\mathbb R^d;\mu),L_2(\mathbb R^d;\mu))\asymp n^{-r_\lambda}(\log n)^{r_\lambda(d-1)}.$$

This is presented as a consequence expected from the conjectured norm equivalence and would extend the established sampling-width rates to even $\lambda>4$; the source does not state that it has been proved.