Refined Oort analogue for generic supersingular hyperelliptic curves

Let $C$ be a generic supersingular hyperelliptic curve of genus $g$ over a field of characteristic $2$. Refined hyperelliptic Oort conjecture. If $g$ is not a power of $2$, then

$$\operatorname{Aut}(C)=\{\pm 1\}.$$

More strongly, the condition may be expected to be an if and only if: the automorphism group is $\{\pm 1\}$ exactly when $g$ is not a power of $2$. This refinement is motivated by the counterexamples in genera $2,4,8$; its general validity is left open.