Conjecture on superspecial hyperelliptic curves of genus 4 with dihedral automorphisms

Let $H$ be a hyperelliptic curve of genus $4$ over a field of characteristic $p$, and let $\mathrm{Aut}(H)$ denote its automorphism group. A curve is superspecial when its Jacobian is isomorphic over an algebraic closure to a product of supersingular elliptic curves. Write $\mathbf{D}_n$ for the dihedral group of order $2n$.

Superspecial genus-4 dihedral automorphism conjecture. The following statements hold:

1. For every characteristic $p\geq 19$ with $p\neq 41$, there exists a superspecial hyperelliptic curve $H$ of genus $4$ such that

$$\mathrm{Aut}(H)\cong \mathbf{D}_4.$$

2. If $p\equiv 3\pmod 8$, there does not exist any superspecial hyperelliptic curve $H$ of genus $4$ such that

$$\mathrm{Aut}(H)\supset \mathbf{D}_8.$$

3. If $p\equiv 3,7\pmod {10}$, there does not exist any superspecial hyperelliptic curve $H$ of genus $4$ such that

$$\mathrm{Aut}(H)\supset \mathbf{D}_{10}.$$

These claims summarize computational search results for primes in a large range. The existence assertion and the two nonexistence assertions are presented as conjectural extensions of those computations to arbitrary characteristics satisfying the stated conditions.