The group-isomorphism conjecture for superelliptic Jacobians over finite fields

Let $C$ be a superelliptic curve given by

$$y^q=F(x),$$

where $q$ is prime and $F\in\mathbb F_p[x]$ has degree not divisible by $q$. Let $J$ be the Jacobian of $C$, and set $k=\operatorname{ord}_q p$.

Superelliptic Jacobian group-isomorphism conjecture. There exists an isomorphism of abstract groups

$$J(\mathbb F_{p^k})\cong J(\mathbb F_p)^k.$$

The statement is presented as being supported by numerical evidence for superelliptic curves; the supplied text gives no resolution, so its status remains open.