Dolgachev's essential-dimension and Cremona-dimension inequality

Let $G$ be a finite group over the complex numbers. Write $\operatorname{ed}(G)$ for the essential dimension of $G$, and $\operatorname{Crdim}(G)$ for the least integer $n$ such that $G$ embeds into the Cremona group $\operatorname{Cr}(n)$.

Dolgachev's inequality.

$$\operatorname{ed}(G)\geqslant\operatorname{Crdim}(G).$$

The source attributes this claim to I. Dolgachev as unpublished. It compares the minimal dimension of a versal faithful $G$-variety with the minimal dimension of affine space on which $G$ acts birationally; the source gives no resolution status.