Consider an algebraically closed field $k$ of prime characteristic. It is widely known that up to $k$-isomorphism there is a finite number of supersingular elliptic curves over $k$. Let $C$ be a projective reduced irreducible smooth genus 2 curve over $k$. It is called *supersingular* if the Jacobian $J_C$ is supersingular.

Is there a finite number of supersingular genus 2 curves over $k$ up to $k$-isomorphism?