Non-finite type étale sites over fields

We consider the notion of finite type-ness of a site introduced by Morel and Voevodsky, for the étale site of a field. For a given field $k$, we conjecture that the étale site of $Sm\mathrm{/}k$ is of finite type if and only if the field $k$ admits a finite extension of finite cohomological dimension. We prove this conjecture in some cases, e.g. in the case when $k$ is countable, or in the case when the $p$-cohomological dimension $c{d}_p(k)$ is infinite for infinitely many primes $p$.