Pro-étale locally simply connected schemes

Question

In topology, topological manifolds are locally simply connected. However, in the étale topology of schemes, the analogue statement is not true: If $k$ is a field then finite separable field extensions $k^{sep}/l/k$ are cofinite in the étale coverings, and we have $$\pi_1^{ét}(\text{Spec}(l),k^{sep})=\text{Gal}(k^{sep}/l)$$ and thus unless $[k^{sep}:k]$ is finite, $\text{Spec}(k)$ is not étale locally simply connected. However, the field extension $k^{sep}/k$ is pro-étale, and thus in the pro-étale topology, $\text{Spec}(k)$ is simply connected. Hence I'm wondering if every scheme of finite type over a field $k$ is pro-étale locally simply connected.