Malle's prediction for twist classes of genus two curves with dihedral automorphisms

Let $K$ be a number field, let $C$ be a genus two curve over $K$, and let ${\mathcal M}_2(K)$ denote the set of genus two curves over $K$. Write $\operatorname{Ht}(C')$ for the height defined in the source, and let $D_4$ be the dihedral group of order eight.

Malle's prediction. If $\operatorname{Aut}_{\overline{K}}(C)\cong D_4$, then

$$\#\{C'\in {\mathcal M}_2(K): \operatorname{Ht}(C')\leq B \text{ and } C'\cong C \text{ over } \overline{K}\}\sim B^{1/4}\log(B)^2.$$

This is a prediction obtained from Malle's conjecture for $D_4$-octic extensions, translating the expected discriminant asymptotics into a count of curves in the twist class of $C$. The source does not state that this prediction has been proved.