Let $k$ be an algebraically closed field of characteristic zero. Let $G$ be a group algebraic space over $k$ such that $G\to $ Spec $k$ is locally of finite type.

Suppose that $G(k)$ is finite.

Does it follow that $G\to$ Spec $k$ is finite (and thus $G$ is a scheme)?

If we assume $G$ is quasi-separated over $k$, then $G$ is in fact a scheme (see Commutative group algebraic spaces). It follows then that $G$ is finite.

But what if $G$ is not quasi-separated over $k$? Is $G$ still finite over $k$?