Field-of-definition conjecture for superspecial genus-4 hyperelliptic curves

Let $H_a$ be a genus-4 hyperelliptic curve over an algebraic closure of $\mathbb{F}_p$, with parameter $a$. Field-of-definition conjecture. If $H_a$ is superspecial, then $a\in\mathbb{F}_{p^2}$. The supplied context does not state whether this is known in any broader generality or explain the precise family of curves represented by $H_a$.